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Introduction to p-Adic Numbers and Valuation Theory
George Bachman
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Introduction to
p-Adic Numbers
and
Valuation Theory
*
By George Bachman
POLYTECHNIC INSTITUTE OF BROOKLYN
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Copyright © 1964, by Academic Press Inc.
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Preface
The field of p-adic numbers, which can be obtained from the
field of rational numbers by a completion process with respect
to a special kind of mapping, or valuation, similar in many respects
to the ordinary absolute value mapping has a great number of
interesting properties. However, beyond this, it turns out that
such fields are of particular interest and importance in algebraic
number theory and in algebraic geometry. This is equally true
of the general notion of a valuation together with some of its
related concepts.
The book is meant to serve as an introduction to valuation
theory. The first two chapters have been written mainly for
advanced undergraduate students and first year graduate students.
The amount of algebra required is quite small, and the algebraic
results needed for these two chapters are included in the first
four sections of the appendix. It is hoped that in this fashion
these two chapters will be reasonably self-contained and available
to as wide an audience as possible. In addition, exercises have
been added to these chapters most of which are intended to give
the reader some manipulative facility with the concepts intro-
duced.
The remaining three chapters definitely demand more mathe-
matical maturity on the part of the reader. At least a first course
in modern algebra would be required to read parts of them.
Although most of the material needed for these chapters has been
stated in the appendix, this is meant just to serve as a handy
glossary for the reader. References have been supplied for this
material.
It is hoped that the treatment of the material throughout the
book itself will prove to be leisurely. We have been somewhat
repetitious in places in the hope of gaining increased clarity.
Again, the most elegant proofs of some theorems have not been
used when it was felt that other proofs were clearer. The develop-
ment of certain sections is based to a large extent on lectures
given by E. Artin. In particular, Sections 1, 2, 4, and 5 of
v
vi	Preface
Chapter III are based on lecture notes written by the author for
a course given in algebraic geometry at N.Y.U. by Professor
Artin (see [3] in bibliography). Needless to say, the author
assumes all responsibility for the proofs presented.
In a short introductory treatment of this sort, there was no
opportunity to present the many beautiful applications of the
subject matter to such areas as algebraic number theory or
algebraic geometry. Although some hints of the applications to
number theory are given. In the bibliography, a number of
books which treat the applications have been listed along with
texts and periodical literature which go deeper into the subject
matter itself, and, finally, some books relevant to the material
listed in the appendix have been added. No attempt has been
made for anything resembling a complete bibliography.
We follow the customary abbreviations by writing, for example,
II, (1.3) in referring to Chapter II, Section 1, Equation (1.3).
In referring to the appendix, we write, for example, A,2 meaning
Section 2 of the appendix. Numbers occurring in brackets refer
to the bibliography.
The author would like finally to express his appreciation to
Professor W. Magnus for his encouragement and advice through-
out the writing of this book. He would also like to express his
gratitude to Mr. Lawrence Narici for his helpful suggestions in
the preparation of the manuscript.
G. Bachman
Brooklyn, New York
Contents
PREFACE...................................................... V
SYMBOLS USED IN TEXT ...................................... ix
Chapter I	Valuations of Rank One
1.	p-Adic Valuations of Q............................. 1
2.	Definition of a Valuation of Rank 1................ 4
3.	Equivalent Valuations............................. 16
Exercises......................................... 23
Chapter II	Complete Fields and the Field
of p-Adic Numbers
1.	Completion of a Field with Respect to a
Valuation ............................................ 24
2.	p-Adic Numbers ................................... 33
3.	Some Analysis in Qr............................... 43
4.	Newton’s Method in Complete Fields................ 52
5.	Roots of Unity in Qv.............................. 61
Exercises......................................... 63
Chapter III	Valuation Rings, Places, and
Valuations
1.	Valuation Rings and Places........................ 65
2.	Valuations........................................ 70
3.	Valuations of General Rank........................ 76
4.	The Extension Theorem............................. 83
5.	Integral Closure.................................. 88
Chapter IV	Normed Linear Spaces
1.	Basic Properties of Normed Linear Spaces.......... 92
2.	Linear Functionals ............................... 99
3.	Banach Algebras ..................................110
vii
viii
Contents
Chapter V	Extensions of Valuations
1.	The Extension Problem..............................117
2.	The Number of Extensions of a Valuation........... 128
3.	Valuations of Algebraic Number Fields—
Examples ............................................ 137
4.	Discrete Valuations................................142
Appendix
1.	Sets and Mappings.............................. 153
2.	Number Theory ................................. 155
3.	Groups......................................... 158
4.	Rings, Ideals, and Fields...................... 161
5.	Glossary for Rings and	Fields...................164
6.	Adeles and Ideles.............................. 167
bibliography  .......................................... 168
SUBJECT INDEX............................................171
Symbols Used in Text
The chapter, section, and page on which they are first introduced are
listed.
z Q R C II* d	Set of all integers, I, 1, p. 1. Set of all rational numbers, I, 1, p. 1. Set of all real numbers, I, 1, p. 1. Set of all complex numbers, I, 1, p. 1. p-Adic valuation of Q, I, 1, p. 2. Distance function, I, 1, p. 3, and IV, 1, p. 93.
v(a) — —Iog| a \, I, 2, p. 5.
Hi — II2 К Q» ordp x ord x a < b	Equivalent valuations, I, 3, p. 16. Completion of a field k, II, 1, p. 26. p-Adic numbers, II, 2, p. 34. Ordinal of x at p, II, 3, p. 45. Ordinal of x, V, 4, p. 143. In an order group, III, 2, p. 71.
К* ~ К — {0} Multiplicative group of the field K, III, 2, p. 73.
II II II lli~ II ll2 II lie	Norm mapping, IV, 1, p. 92. Equivalent norms, IV, 1, p. 94. О-Norm on a finite-dimensional vector space, IV, 1, p. 95.
ll/ll a(x) •^X7fc(X) e f	Norm of a bounded linear functional, IV, 2, p. 101. Spectrum of x in a Banach algebra, IV, 3, p. 113. Norm from К to k, V, 1, p. 118 (A, 166). Ramification index, V, 1, p. 121. Residue class degree, V, 1, p. 121.
ix

CHAPTER I
Valuations of Rank One
1. p-Adic Valuations of Q
In this chapter we wish to give a provisional definition of a
valuation, or, more accurately, we wish to define a valuation of
rank one. In Chapter III, this concept will be generalized.
Throughout the text, we shall use the following notations: Z, the
ring of integers; Q, the field of rational numbers; R, the field of
real numbers; and C, the field of complex numbers.
Before giving the definition of a rank one valuation, we wish to
motivate the definition by considering an example. Our aim is to
extend the notion of the ordinary absolute value function on the
field Q. We recall that the basic properties of this function,
| |: Q —► R, are the following:
forxeQ, | x | > 0 and =0 if and only if x = 0;	(1.1)
for x, yeQ, | xy | — | x | | у |;	(1.2)
(the triangle inequality) | x 4- у |	| x | + | у |. (1.3)
We shall now construct other mappings of Q into R which
satisfy these conditions. Actually these functions will satisfy a
much stronger condition than (1.3). Instead of “weighing” the
rational number x according to its numerical value | x |, we shall
now “weigh” it in a certain sense according to the presence of a
fixed prime, p. We now proceed more precisely in constructing
these functions.
Let c e R, where 0 < c < 1; c will be fixed throughout the
discussion. Also, let p be a fixed prime number. If x is any
rational number other than 0, we can write x in the form
a
I
2
I. Valuations of Rank One
where a, b € Z, p f a, and p f b, and where a e Z, Clearly, a may
be positive, negative, or zero depending on x. We now define
| X |„ — c°-, and | 0 |B = 0.
It follows immediately from the definition that | x |„ > 0 and
equals 0 if and only if x = 0. Furthermore, if у = p^fa'lb'),
where p-\ a' and p f b', then
XV = р<*+$ — ,
bb
where p -f bb' andp -f aa'. Hence,
I ХУ |p =	= I x |„ | у |B .
Finally, we shall show that
l*+j|p < max (| x |„ , | у |b).	(1.4)
It is clear that (1.4) is a much stronger statement than (1.3), i.e.,
from (1.4) it follows immediately that the function | |„ satisfies
condition (1.3). Instead of proving (1.4), we shall prove the
following equivalent condition:
I x Ip *4 1 I 1 + x |p 1-	(1-5)
We first note that (1.4) is, indeed, equivalent to (1.5), for, sup-
pose (1.4) is satisfied, then, if | x |„ + 1,
I 1 + x |,„ + max (| x |„ , 1) = 1,
and (1.5) holds. Conversely, suppose (1.5) is true. We may
assume that у 0, for, if у = 0, (1.4) is trivially true. Also, we
may assume without loss of generality that | x |„ + | у |„ . Then
I Х!У Ip 1 and, therefore, by (1.5)
1+ H ^1, so | x +y |„ sg |y = max (| x |„ , | у |„).
1 У
Thus, we must just show that (1.5) is true. It follows immediately
frc^m the definition of | x |p for x + 0 that if | x |s + 1, then
1. p-Adic Valuations of Q	3
a 0, and x can be written in the form x — cjd, where (c, d) = 1,
and p -Г d. Then
and 1 + x also has a denominator prime to p\ whence,
| 1 + x |„	1. Since (1.5) holds trivially if x — 0, the proof is
completed.
We have, therefore, shown that the function | |„ satisfies all the
conditions that the ordinary absolute value function satisfies, and,
moreover, | satisfies condition (1.4), which is certainly not
true for | |. We wish to make some further observations concerning
this function. Define in terms of the function | ]„ a new function,
d-. QxQ^R,
namely,
d{x,y) = | x -y |„.	(1.6)
Let us see what properties the function d possesses. Clearly
d(x, y) 0 and =0 if and only if x=y. (1.7)
<Z(y,x) = I у -x|„ = I -1 |„ I X —y = |x-y |„ = d(x,y\
so
d(y, x) = d(x, y).	(1.8)
Finally,
d(x, г) = | x - z |„ = | (x — y) + (y - z) |„
max(| x —у , |y — z |„)
— max (d(x, y), d(y, г)).
Thus,
d(x, z) max (d(x,y), d(y, z)).	(1.9)
From (1.9), it follows immediately that
d(x, z) d(x, y) + d(y, z).	(1-Ю)
A set X together with a function, d\ X x X —► R, which
satisfies the conditions (1.7), (1.8), and (1.10) is called a metric
4
I. Valuations of Rank One
space. Hence, Q together with the function d(x, у) = | x — у is
a metric space, which actually satisfies the stronger condition
(1.9). Equation (1.9) is frequently called the ultrametric inequality.
In a metric space, W, one can introduce the notion of conver-
gence of a sequence of elements. There are a number of equivalent
ways of introducing this concept, but the following will suffice
for our purposes. The sequence {x„} in the metric space X is said
to converge to the element x e X, written xn —> x, if and only if
d(x, x„) —> 0, as n —> oo.
Let us now observe that in the metric space Q introduced
above, the sequence
p, p2, ...,pn,...
converges to 0; namely,
40, ^«) = |/1«|₽ = cn,
and, since 0 < c < 1, cn —► 0 as я
At first sight, this might seem strange, but when one recalls
that the distance function was defined in a manner entirely
differently from the customary absolute value, this behavior of the
sequence {pn} should not seem too surprising.
It is not difficult to show (see Exercise 1, page 23) that if X is a
metric space in which the distance function d satisfies the ultra-
metric inequality, then, if d(x, у) Ф d(y, z),
d(x, z) = max (d(x, y), d(y, z)).
This result has the following geometric interpretation: every
triangle in such a metric space is isosceles, and its base has length
less than or equal to that of the equal sides.
2. Definition of a Valuation of Rank 1
We shall now extend the concept of the absolute value function
on the field Q; namely:
2. Definition of a Valuation of Rank 1
5
Definition 2.1. A valuation of rank 1 of a field A is a mapping,
| |, from k into R such that for all a, b e k
| a | > 0 and =0 if and only if a = 0;	(2Л)
l«N - l«I IN;	(2.2)
| a + b |	| a | + | b j.	(2.3)
We shall generally omit the expression “of rank 1” in the first
two chapters, and simply speak of such a function as a valuation,
it being understood that such a map is always into R.
If the valuation satisfies, in addition, the condition
| a 4- b | max (| a |, | b |)	(2.4)
then it will be called a non-archimedian valuation. From the results
obtained in Section 1, we see that the function | |„ is a non-
archimedian valuation of Q. We shall shortly see that there are
essentially no other non-archimedian valuations of Q except for a
trivial one. The valuation | |„ is called the p-adic valuation of Q.
Before discussing various results which follow from the
definition of a valuation, we shall reformulate the definition of a
non-archimedian valuation. For a e k, and a 4 0, we define the
function v(a) as follows:
w(a) = —log | a |,	(2.5)
while
w(0) = °°,
where we have introduced the symbol °° and shall operate with it
in the customary formal fashion. Thus, v is a mapping of k into
the extended real number system, i.e., v(a) e R v {°0}, and
v(a) = °° if and only if a = 0. Let us see what further conditions
v satisfies. Since | ab | = | a | | b |, we get
log I «N = log | a | | b | = log | a | 4- log | b |,
-log | ab | = -log | a | — log | b |.
This, together with the rules for operating with <», gives
v(ab) = v(a) 4- v(b).	(2.6)
6
I. Valuations of Rank One
Also, since | a 4- b | max (| a |, | b |),
log | a + b | < max (log | a |, log | b |),
and
—log | a 4- b | —max (log | a |, log | b |),
i.e.,
—log | a + b | > min (—log | a —log | b |),
and we have
v(a + b) > min (v(a), w(6)).
(2-7)
Thus, the function v satisfies the conditions (2.6) and (2.7)
analogous to the conditions (2.2) and (2.4) for the non-archi-
median valuation | |.
The following fact, which we formalize as a theorem is fre-
quently useful—particularly in the consideration of examples.
Theorem 2.1. Suppose | |T is a mapping of an integral domain
A into R which satisfies the conditions (2.1) to (2.3). Then | jj
can be extended uniquely to a valuation, | |, of the quotient
field k.
Proof. If there exists a valuation | | on k extending | , then it
is clear that for x e k, where x = alb, a, be A, we must have
„ I I a I _ I a li
1 ~ l ь ।	। ь |г 
(2-8)
Thus we have uniqueness. Now let us show that (2.8), indeed,
defines an extension | | of j |г which is a valuation. We must first of
all show that j | is well defined. Thus, suppose a/Z> = c/d, then
ad = be, and | ad = | be ,
or
| a j d |^ — j b | c ,
I a li __ I c li
l*li I d I, ‘
2. Definition of a Valuation of Rank 1
7
Now we show that | | satisfies the conditions for a valuation.
Condition (2.1) is clear as is (2.2). If у — ejd> c, de A, then
and
.	,	,	I ad + be L _ I ad I, , j be L
1 x + y 1 |M|X ГЖ + ГЖ
_ Ia li । ic li
Ш1 Hli
= I X I + |j |.
Finally, it is clear that | | extends | |г for if a e A, then a = ab/b,
where be A, and
which completes the proof.
It follows immediately from Definition 2.1 that a valuation | |
satisfies the following conditions
I 1 I = 1
1-11 = 1
| a | _ | a |
I b I ТИ
||«|- 1&Ц \a — b\.
(2-9)
(2.Ю)
(2.П)
(2.12)
Let us now see what the counterpart of (2.12) is for a non-
archimedian valuation. Thus, suppose | | is non-archimedian, and
suppose | a | > | b Then
| a | = | {a + b) — b | max (| a 4- b |, | b |).
Clearly | a 4- b | must be the maximum, for, otherwise, we would
have I a | < | b |, which contradicts our assumption. Hence,
I a j < j a + b I < max (j a | b |) = | a (.
8
I. Valuations of Rank One
We, therefore, have:
Theorem 2.2	. If | | is a non-archimedian valuation, and if
| a | > | b |, then | a -f- b | ~ | a |.
It is easily seen, by taking a — —b, that the theorem is not true
if | a | = | b |. It also follows readily that, for a non-archimedian
valuation,
I «1 + «2 +----h a» | max (| «J |, | a2 |,	| an |),
and
I «I + a2 + ••• + «»! = I «1 | if |	\ < | «1 |
for j — 2, n.
We continue to assume that | | is a non-archimedian valuation,
and we consider the set V of all a e k such that | a |	1. If a,
beV, then
I | = | a | | b |	1,
and, therefore, ab e V. Also,
J a — b | max (| a |, | b |)	1.
Hence, V is a ring; also 1 e V.
Next, we consider P, which is the set of all a e k such that
| a | < 1. Clearly, if a, b e P, then a + b e P, and if a e P and
b e V, then
| ba | = | b | | a | < 1,
so ba e P. Therefore, P is an ideal in V. Moreover, if a e V and
if а ф P, then | a | = 1. Therefore, | a-1 | =1. It follows that
P is the unique maximal ideal of V, and, since 1 e V, that P is
also a prime ideal. Summarizing, we have:
Theorem 2.3	. If | | is a non-archimedian valuation, then the
set V C k of all elements a such that | a |	1 is a ring with
identity. The set P of all elements a such that | a | < 1 is the
unique maximal ideal of V, and P is also a prime ideal.
2. Definition of a Valuation of Rank 1
9
The ring V is called the valuation ring associated with the
non-archimedian valuation | The field V]P is called the
associated residue class field.
We have already obtained a number of valuations of Q,
namely, the />-adic valuations as well as the usual absolute value.
In addition, there is, of course, the trivial valuation which one
can define for any field k, namely, | 0 | = 0 and | a | = 1 for all
a 0, a e k. We wish to show that these exhaust all possible
valuations of Q in a certain sense; that is, we set forth the problem:
to find all valuations of Q. Before embarking on this project,
we establish the following lemmas which will be needed in the
course of our investigation.
Lemma 2.1. If 0 < r C $, and if at, i = 1,..., n are non-
negative real numbers, then
Proof. Let d — atT. Then
...
d1'' W d’M
1=1
(gm’T
since af/d 1, and since r s. But
Therefore,
10
I. Valuations of Rank One
Lemma 2.2. If 0 < a 1, and if , i — 1, n are non-
negative real numbers, then
(61 + fe2 +	+ ya < bf + у + - + у. (2.13)
Proof. In Lemma 2.1, choose a, = b,, i — 1, ..., n, s = 1 and
r = ol. Then we obtain
+ bs + ••• + bn (У + у + ... + V)V«,
or
(Z>2 + b2 + ••• + b„y* bf + у + - + b„«.
Now, we return to the problem at hand. Let m and n be two
integers >1, and write m using the base n. Then
m = a0 + агп -f- a2n2 -f- ••• + aknk,
where 0	n — 1, i = 0, 1, ..., A. Clearly, nk m. Hence,
k < 1оёот
" logn '
Assume now that | | is a valuation of Q. Then applying (2.3), we
have
| ai I — ! 1 +1 + "• + 1 | П.
Thus,
I m |	| a0 | + | Д1 | | n | + | <z2 j | n |2 +-1- | ak | | n |*
< n(I + | n | + | n (2 4-	+ | n I*)
n{k 1) max (1, | n |)fc.
Using the estimate on k, we, therefore, have
| m | n 4- 1) max (1, | n |)iogm/iogn> (2.14)
which is true for any m, n > 1. Hence, replacing m by mT in
(2.14), and then taking the rth root of both sides, we get
| m | sg	+ 1) max (1, [ n |)log»»/logn_
2. Definition of a Valuation of Rank 1
11
Finally, if we let r —>	and use the fact that Нтт_мод/т = 1,
then
| m | max (1, | n |)iogm/iognt	(2.15)
There are now two distinct possibilities which we have to
consider. The first is that there exists an n > 1 such that
| я | C 1. Equation (2.15) then yields that | m j 1 for all
m > 1. From which it follows, using the fact that | —1 | = 1,
that | m | <1 1 for all m e Z. In this case, we show that the
valuation, | |, must satisfy the stronger inequality (2.4). Actually,
we can prove a stronger statement, and, because of its importance,
we single this out as a theorem.
Theorem 2.4	. Suppose A is a field with a valuation, j |, and
suppose | m | d for all integers m of k (here the set of integers
refers to the isomorphic image of Z if k has characteristic 0, or
to the isomorphic image of the residue classes modulo p if k has
characteristic/»). Then the valuation, | |, is non-archimedian.
Proof.
| a + b I1 = |(« + i)T |
= j aT + (j) а'-Ч) + Q a*-252 + ••• + 611
< I a |T +	| a I7-1 | b | + ••• + | b |T
(r + 1)<Z max (| a |, | b |)T.
Taking the rth root of both sides and letting r we obtain
| a b | max (| a |, | b |),
which completes the proof.
Thus in our special case of a valuation | | on Q under the first
possibility, we obtain that | | is non-archimedian.
If | m | = 1 for all m 6 Z, m 0, then using Theorem 2.1, we
see that | | is just the trivial valuation. Thus, let us suppose that
12
I. Valuations of Rank One
j m | < 1 occurs for some m > 1. Then let p be the smallest
positive integer such that | p | < 1. We claim that p is a prime,
for if p — ab, where a, b < p, and where a and b are positive, then
which is a contradiction. Hence, p is a prime. Suppose | m | < 1.
Then
m — qp 4- r, where 0 C r < p.
If r 0, then, since r < p, we have | r | = 1. But
I ЧР I = |	+	| C | P | <1,
so | m | = 1, which is a contradiction. Thus, p | m, and, conver-
sely, if | m, then clearly | m | <1.
Now if x G Q, x 0, then x has the form
x = pa^,
where p -Г a and plb, and, therefore, by the immediately
preceding discussion, we have
I X | = | p = ca,
where c = | p | is some real number such that 0 < c < 1. Thus,
in this case, | | is ap-adic valuation. We have seen in Section 1 that
all such functions are, indeed, valuations (non-archimedian) of Q.
Now let us consider the second possibility, namely, for any
n > 1, | n | > 1. It follows from (2.15) that
| m |l/logm | n jl/logn,
and, since this is true for all m, n > 1, we get, interchanging m
and n, the inequality in the opposite direction. Hence,
| m |l/logm = | n |l/10gn. that jS) | m |l/10gm =
where c is independent of m. We observe that c > 1, and,
therefore, we can write c = e“, where a > 0. Then
I m I = e»10gm = ma
(2.16)
2. Definition of a Valuation of Rank 1
13
We can also obtain an upper bound on a, for take m = 2. Since
j 2 | = | 1 + 1 | < 2,
we see that 2“ < 2, so a 1. Finally, let | |да denote the usual
absolute value function. Then for m > 1, we have, using (2.16),
I m I — ma = I m Iе ,
II	' ’co ’
(2.17)
but | —1 | = 1 and | —1	— 1, so (2.17) is true for all meZ,
since it is also trivially true for m = 0. Applying Theorem 2.1,
we have, for any x e Q,
| x | — | x |“ , where 0 < a C 1.
We shall now show that all functions of the form | |“ , where
0 < a 1, and | („о still denotes the usual absolute value func-
tion, yield valuations of Q. Clearly conditions (2.1) and (2.2) are
satisfied. To verify (2.3), we use Lemma 2.2; namely,
I X + у 1“ < (I X |И + IJ |J“ c I X I» + \y IX •
Thus, we have completely solved the problem originally set
forth. We have shown that any valuation of Q is either the trivial
valuation, a />-adic valuation, or a power, | |£, of the ordinary
absolute value, where 0 < a 1.
We shall show that all valuations of the form | |X , 0 < a 1,
are equivalent in a sense which will presently be made precise.
We also note that in defining the />-adic valuation | |„ we chose a
number c, 0 < c < 1. If one chooses another real number d,
0 < d < 1, instead of c, the resulting />-adic valuation will be
equivalent to the original one, again, in a sense which we shall
make precise. Before proceeding, however, to this notion of
equivalent valuations, we wish to consider some further examples
of valuations.
We shall now consider the field k — F(x) of rational functions
with coefficients from a field F. We wish to determine all valua-
tions of k which are trivial on F. It follows immediately from
Theorem 2.4 that any such valuation, | |, must be non-archi-
14
I. Valuations of Rank One
median. As in the case with Q, there are two possibilities to
consider. First, | x |	1. Then, since | | is non-archimedian,
and trivial onF, |/(x) | < 1 for all/(x) gF[x], If |/(x) | = 1 for
all /(x)eF[x], other than 0, then, using Theorem 2.1, it is clear
that | | is the trivial valuation. Thus, let us suppose that
|/(x) | < 1 for some /(x)eF[x]. We know, by Theorem 2.3,
that the set of all such polynomials is a maximal ideal, and since
F[x] is a euclidean ring, this ideal must be a principal ideal
generated by an irreducible polynomial p{x'). One also shows, as
was done in the analogous case for Q, that | f(x) | < 1 if and only
if p(x) |/(x). Now, if h(x)eF(x), write
h(x) = p(x)a ,
и\л/
where a(x), 6(x)eF[x], and p(x) f a(x)t p(x) -Г Then
I Kx) I = I P(x) 1“ =
(2.18)
where c ~ | p(x) |, and 0 < c < 1. One also shows, as in the case
for Q, that all such functions of the form (2.18) yield valuations
of F(x).
For the second possibility, we consider | x | > 1. Let
/(») = anxn +	+------h aAx + a0 •
Then |/(x) | = | x I" — | x |deg/<®> = cdeg/(®)) where c = | x |,
and c > 1. Thus, if h(x) eF(x), A(x) — f(x)/g(x), where /(x),
g(x) eF[x], and
I h(x) I =	! = I x |degr-deg9 _ ^egr-flegg. (2.19)
I I
where c > 1. Again, it is easy to see that all functions of the form
(2.19) yield valuations of F(x).
As a natural extension of the />-adic valuations of Q and the
case (2.18) forF(x), we consider A to be any unique factorization
domain. Then, if x e A,
X = e
2. Definition of a Valuation of Rank 1
15
where e is a unit, and all but a finite number of the nonnegative
integers = 0, and then’s are irreducible elements of A, which
are not associate elements. We then define
I x i; = л,
(2.20)
where 0 < c < 1. The functions (2.20) satisfy conditions
(2.1)—(2.3) on A, and are extended uniquely to the quotient field k
by Theorem 2.1. All such functions are valuations of k, and
(2.20) is called the p-adic valuation of k.
If | | is a valuation of k, we can view k as a metric space with
distance function, d(x, yi), x,yek, defined as
d(x, j>) — | x — у |.
It follows immediately from the definition of valuation that this is,
indeed, a legitimate distance function; that is,
d(x, У) 0 and =0, if and only if x = y;
d(x,y) = d(y, x);
d(x, z) d(x, y) -|- d(y, z).
If, in addition, | | is non-archimedian, then d will satisfy the
ultrametric inequality:
d(x, z) "C max (d(x, y), d(y, z)).
The proof is like the one in the first section for | |„ .
Once we have a metric space, we can introduce neighborhoods:
S(x, r) = {y e k | d(x, y) < r}.
S(x, r) is called the r neighborhood of x, or the spherical neigh-
borhood of x with radius r. In terms of these neighborhoods,
one can now introduce the usual topological notions of open set,
closed set, compact set, convergence, etc.
16
I. Valuations of Rank One
3.	Equivalent Valuations
Let | | be a valuation on the field k. Also, let {an} be a sequence
of elements of k.
Definition 3.1. The sequence {an} is called a Cauchy sequence
with respect to the valuation | | if, for any real number e > 0,
there exists an integer W such that | an — am | < e for all n,
m > N.
In terms of the metric introduced at the end of Section 2,
we could express this by writing d(an , am) < e for all n,
m> N.
Definition 3.2. The sequence {«„} is called a null sequence with
respect to the valuation | | if, for any real number e > 0, there
exists an integer N such that | an | < e for all n > N.
Again, in terms of the metric d, we could express this as
d(0, a„) < e for n > N.
Now, let | and | |2 be two nontrivial valuations of k.
Definition 3.3. The nontrivial valuations | |x and | |8 are
called equivalent, and we write | |x ~ | |2 , if j a |x < 1 implies
1 л |2 < 1-
Clearly, if every null sequence with respect to | |x is a null
sequence with respect to | |2, then | |x ~ | |2, for suppose
| a < 1, then the sequence {a”} is a null sequence with respect
to | |x . Therefore, it is a null sequence with respect to | |2 , so we
must have | a |a < 1.
We shall now investigate the relation between valuations.
Clearly, ~ is a reflexive and transitive relation. We shall pre-
sently see that it is also a symmetric relation, and, hence, is an
equivalence relation. First, note that if | |x ~ | |2, and if
| a > 1, then | a |2 > 1, for | а > 1 implies | 1/a < 1.
Therefore, j 1/a |2 < 1, or [ a |2 > 1.
Theorem 3.1	. If | |x ~ | |2, then | « |x = 1 implies that
\ a |2 = 1.
3. Equivalent Valuations
17
Proof. Since | |г is not the trivial valuation, there exists an
element b g k, b Ф 0, such that i b |j < 1. Then
I anb |i = | a|rfl I 6 li < 1>
which implies that
| anb |2 < 1, or
/ 1 x1/»
1й1г<\|Тг)
'I о 12'
If we let n —► oo, then we get | a |2	1. But if in place of a we
use Ila, then we get | 1/a (2	1, or | a |2	1. Hence, | a |2 — 1.
It now follows that ~ is, indeed, a symmetric relation, for
suppose j ~ j j2 , and suppose | a j2 < 1. Then if | а |г > 1, we
have | a |2 > 1 by the comment just preceding this theorem.
If | а |г = 1, then | a |2 = 1 by the theorem. Thus, we must
have | a < 1, i.e., | |2 ~ | |i.
We next wish to show that if | |x ~ | |2 , then | |2 is just a
power of j |i. Once this statement has been established, then it is
clear that if |	~ | |2, any null sequence with respect to j is
also a null sequence with respect to | |a , which, together with the
comment following Definition 3.3, would yield an alternate
characterization of equivalent valuations.
Theorem 3.2	. If | |i ~ Ik > then | |2 = I li, where v is a
positive real number.
Proof. We choose a fixed b G k such that | b |x > 1. Let a G k,
and а Ф 0. Then
I a L = | b |i“, where a =	,-т-г •
log | 6 |i
Now, let n, m G Z and n/m > a. Then
I am I
or
which implies, since | |i ~ | |2 , that
j bn
1.
2
18
I. Valuations of Rank One
Therefore,
I a l8 < I l”/m.
In a similar fashion, one gets if nfm < a, then
> I b I"/"1.
2 I u '2
Hence, we must have
I a Is — I la*
Thus,
log I a lx = log I a la
log I b |x log | b |s ’
so
, ii log | b I» . . ,
log a 2 — . - ". • ~ ’ log a , ,
81	2 log | Hl ё 11
and if we let
log I b Is
log I b |i ’
then
| a |s = | a If.
Since the result is trivially true for a = 0, the proof of the
theorem has been completed.
We next establish the important approximation theorem, which
pertains to a finite number of inequivalent valuations. Before
stating and proving this theorem, we establish two useful lemmas.
Lemma 3.1. Given n inequivalent (nontrivial, as usual)
valuations | |f , i = 1, ..., n on a field k, there exists an element
a G k such that | а |г > I and | a < I, i — 2, ..., n.
Proof. First, let n = 2. Then there exists an element b 6 k
such that
I b |i > 1, and I b |2	1
3. Equivalent Valuations	19
since | |j and | ]2 are not equivalent. Also, there exists an element
c 6 k such that
| c |j < 1, and | c |2 > 1
for the same reason. Now, let a = blc. Then
We now assume that the lemma is true for a — 1 inequivalent
valuations and proceed to the case of n inequivalent valuations.
By the induction hypothesis, there exists an element d. e k such
that
| d |i > 1, and | d ], < 1 for i — 2, ..., n — 1.
Also, there exists an element c e k such that
| c |i > 1, and | c |n < 1
I
since | |j And | |n are inequivalent and since the lemma has been
established for n = 2. Suppose first that | d 1. Then let
a = djc withy a sufficiently large positive integer:
| a |i = | d l^’] с > 1, И„ = ИИ‘|П<1.
and, for 2 < i < n,
I a = | d |г’ | c |,- < 1
if j is sufficiently large. Therefore, in this case, the element
a — djc for j sufficiently large works.
If, however, | d |n > 1 let
<Z’c
a ~ ‘
Then since
\ d> I _ I 1 | . I (W |	.
I 1 + I 1 + к 11+ (l/rf); к
20
I. Valuations of Rank One
we have that
djc
It follows by (2.12) that | a |x is arbitrarily close to | с |г for j
sufficiently large. Thus, | а |r > 1 for j sufficiently large. Now,
and, as above, the right-hand side of this inequality approaches
| c |n as j gets large. Hence, | a |n < 1 if j is sufficiently large.
Finally, if 2 i < n, then
1 '•	1 - ldl,<’
and the right-hand side of this inequality is arbitrarily small for j
sufficiently large. Thus, in this case, a = d’c/l + dj for j
sufficiently large satisfies the conditions of the lemma.
Lemma 3.2. Let | |<, i = 1,..., n, be n (nontrivial) inequiva-
lent valuations of a field k and let e be an arbitrary positive real
number. Then there exists an element d e k such that
| d — 1 |i < e, and | d |f < e for i = 2, ..., n.
Proof. We choose an a e k which satisfies the conclusion of
Lemma 3.1, and set
d = —?L—.
1
Then
l'-n.4T^L<T^T<'
for j sufficiently large, where e is any arbitrarily preassigned
positive real number. Next, for 2 i n,
for j sufficiently large, and the proof of the lemma is completed.
3. Equivalent Valuations	21
Theorem 3.3 (the approximation theorem). Given n inequi-
valent (nontrivial) valuations | |,, i = 1,..., n, of k and n arbitrary
elements at, i — 1, n of k, there exists an element a e k such
that
(: = 1...я),
where < is an arbitrarily given positive real number.
Proof. Lemma 3.2 guarantees that there exist elements
d{ e k such that
|	— 1 |f < < and | dt |, < e' for j ф i.
Let
a —- a1dl -J- a2d2 + ” • + andn .
Then
I a ~ ai I» = I el<A + ”* + ai{di — 1) + ” + It
I al If * +	+ I ai |i e> 4- "‘ + | an |t
< €,
for e' sufficiently small, and the proof has been completed.
In the case of k = Q and with | |< = | |₽i as/>f-adic valuations,
one can establish in a simpler fashion a much stronger version of
the approximation theorem: not only does there exist an a<=Q
such that | a — at |„ < e (i = 1,..., я), but | a.\v C 1 for all
other />-adic valuations. We shall now establish this contention.
First of all, we claim that it suffices to prove the statement where
all elements involved are integers. Thus, given integers c,, and
distinct primes />f (i = 1,..., яг), and 8 > 0, suppose we can
find an integer c such that
| c — c{ |y. < 8 (г = 1, ..., tn), and | с |„ С 1
for all other primes p, then we show that the original problem has
a solution; namely, let d be a common denominator of the
and take
Ci = dat (i = 1,..., n), and =0 (г = n 4- 1,..., яг),
22
I. Valuations of Rank One
where m — n equals the number of primes p for which | d |p ф 1,
and, finally, let
8, — min I d I- e, 8, = min I d L ,
where the minimum for 82 is taken over those p for which
|	1. Take
8 = min (8г, 82).
Then there exists an integer c such that
I c — da.i |P( < 8 < \d |p.« (i = 1,..., n)
| c - 0 |„ < 8 < | d |„
for those p such that | d |P ]; and
I c Io 1 = I I®
at all other primes. Thus, we have an integer c such that
j c — dat I,. < | d |„. c (i = 1,..., n),
and
at all other primes. Then, clearly, a — cld solves the original
problem.
Now the condition | a |„	1 for an integer is automatically
satisfied, so we must just see if we can satisfy the condition
la-ad₽(<« (t =
This means
a = a-i (mod p?*)	(: = 1,	n),
with пг sufficiently large. Since the moduli are relatively prime,
one knows that these simultaneous congruences can be solved by
the Chinese remainder theorem, (see Appendix) which completes
the proof.
Exercises
23
Exercises
1.	Let the set X together with d be a metric space. Show that if
d satisfies the ultrametric inequality, and if d(x, у) Ф d(y, z), then
d(x, z) — max (d(x, y), d(y, z).
2.	In the definition of the p-adic valuation, | |„ , take c = 1/p
(the normalized p-adic valuation). Show that if x e Q and x 0,
then
where the product is taken over all primes p, and where |
denotes the usual absolute value.
3.	In a metric space X with distance function d, let Sr(x) be the
spherical neighborhood of x with radius r. Show that if d satisfies
the ultrametric inequality, and if z G Sr(x), then Sr(x) — Sr(z),
i.e., every point inside a sphere is a center.
4.	Let V be the valuation ring associated with the non-archi-
median valuation | |. Let U be the set of all ae V such that
| a | = 1. Show that U is a group (called the group of units of F).
5.	Prove that any valuation of a field of characteristic not equal 0
is non-archimedian.
6.	Prove that the only valuation of a finite field is the trivial
valuation.
7.	Show that the nontrivial valuations | |г and | |2 are equivalent if
and only if every Cauchy sequence with respect to | |x is a Cauchy
sequence with respect to | |8 .
8.	Assuming the stronger version of the approximation theorem
in Q, prove the Chinese remainder theorem.
9.	If | | is a non-archimedian valuation of a field k, show that
{x„} is a Cauchy sequence with respect to | | if and only if
I *n+i — xn I -> 0 as n <».
10.	Let A be a field and | | a valuation of k. Show that the field
operations are continuous with respect to | |.
CHAPTER II
Complete Fields and the Field of
p-Adic Numbers
1. Completion of a Field with Respect to a Valuation
As the reader is well aware from analysis and algebra, there are
great advantages in passing from the field Q to the field R. One
shortcoming of Q is the fact that not every Cauchy sequence of
rational numbers converges to a rational number. These “gaps”
in the field Q are filled in by a completion process which yields the
field R. One way of achieving this completion is by Cantor’s
method, and it is this technique which will be applied to the
more general situation which will confront us of an incomplete
field with respect to a valuation. We must first, however, introduce
some terminology.
We have already defined in I, 3 a Cauchy sequence {o„} of
elements of a field k with respect to a valuation | | in k.
Definition 1.1. Let k be a field and | | a valuation on k. Let
be a sequence of elements of k. The sequence {an} is said to
converge to the element a e k (and a is said to be a limit of {an},
denoted by lim an ~ a, or an —> a) if, for every real number
e > 0, there exists an integer N such that | an — a | < e for
all n > N.
In terms of the related metric of I, 2, this means d(an , a) -> 0
as n —>°°.
The customary statements concerning limits which the reader
is familiar with for real and complex numbers hold in our more
general situation since the valuation satisfies the same conditions
that the usual absolute value function does and which are needed
in proving those statements.
Definition 1.2. The field k is called complete with respect to
the valuation | | if every Cauchy sequence of k with respect to | |
has a limit in k.
24
1. Completion of a Field
25
As noted, the field Q is not complete with respect to the usual
absolute value, whereas, R and C are both complete with respect
to the customary absolute values.
Before considering the general process of completing a field, we
shall establish two useful results which hold in complete fields
with respect to a valuation. If A is a complete field with respect
to a valuation | |, we can introduce the notion of convergence
of an infinite series an , in the usual fashion. It follows
immediately that if S“=1 an converges, then lim — 0. It is
useful to note that if | | is non-archimedian, the converse
statement holds, namely:
Theorem 1.1. If k is a complete field with respect to a non-
archimedian valuation | |, and if {a*} is a sequence of elements of
k such that lim an = 0, then an converges.
Proof. Let
4* *'' 4~	4“ '’' 4~	>
where m < n. Then
I I = I «n + ’•• + a»n+l I	J ai I -* 0
as n, m —>-	which implies that lim sn exists since k is complete.
Therefore, S„=i an converges.
Theorem 1.2. If k is a complete field with respect to a
valuation | |, and if {<zn} is a sequence of elements of k such
that SXi | an | converges, then an converges.
Proof. Let 4 = I ax ) 4-4-I «п I , sn = «i 4-------4- an •
Then, for n > m,
I V — sm I — II an | 4-	4- | am+11|	(1.1)
where | rn' — sm' | refers to the ordinary absolute value on R and
similarly for the outer symbol | | on the right of (1.1) embracing
the sum. But
11 &п I 4- " 4- | ^m+l ll=l<^|4~’“4-| I
26 II. Complete Fields and Field of p-Adic Numbers
since | a, |	0, and
I $n $m I I &n 1 d- ‘" + I &m+l I * 0
as n, m -> oo since S“=1 | an | converges. Thus | sn — sm |
—> 0 as n, m —> °°, and, since k is complete, the theorem follows.
Now, we proceed to a discussion of the completion process.
Given a field k with a valuation | we shall construct a complete
field k, called the completion of k, which contains a field k,
isomorphic to k and such that k is dense in k. Moreover, k is	;
unique up to an isomorphism. Actually a little more is true, not
only is k isomorphic to k, but it is also isometric to k, i.e., the
isomorphism preserves distances also. Also, & is unique up to an
isometric isomorphism, or, more biefly, a congruence. We turn to
the construction.
Let {an} and {An} be two Cauchy sequences of k with respect
to | |. Introduce an addition and a multiplication between Cauchy	>
sequences by defining:
{®n} “1“ {M d- >	{^n}	~
Let us show that the set of all Cauchy sequence is closed with
respect to these operations. Clearly, {an + 6n} is a Cauchy
sequence since
I d~ bn (am d- ^m) I |	| d~ | bm |-
Since any Cauchy sequence is bounded, {anbn} is also a Cauchy
sequence, for
I ^nJ^rn |	| О-п(Ьп Ьт) d~ bm(un &rn) |
'd | Яп | | bn bm | -|- | bm | | an am |
I bn bm | “b 2^2 |	!>
where | an | Kx for all n, and | bn | -C K2 for all n. The con-
tention is now clear. It is also clear that the set A of all Cauchy

1. Completion of a Field
27
sequences of k with respect to these two operations is a ring with
identity element.
Next, let {a„} and {bn} be null sequences. Since
I M I «J +1b
{an} — {bn} is a null sequence. If {c„} is a Cauchy sequence and if
{an} is a null sequence, then
I I %- | |>
where | cn | К for all n. Therefore, {c„} {an} is a null sequence,
and it follows that the set M of all null sequences is an ideal in the
ring A. We wish to show that M is actually a maximal ideal.
First, observe that if the Cauchy sequence {a^ is not a null
sequence, then there is an e > 0 and an integer N such that
| an | e for n > N.
(1.2)
For otherwise, for every e > 0 and every integer N, there
exists an n > N such that | an | < e. But there exists an integer
N such that
i am ~ °ni < e for n, m > N;
then
for every e > 0 provided m > N, which implies, contrary to
our assumption, that {an} is a null sequence, and, therefore, (1.2)
has been established.
We continue to assume that{an} is a Cauchy sequence which is
not a null sequence so (1.2) holds. Define
for
n < N
for
n > N.
(1-3)
28 II. Complete Fields and Field of p-Adic Numbers
{/>„} is a Cauchy sequence, for
I A _ A I —	1	1
I "n Ощ I —
an am
__ I &n I
I	I
<" । й»»	।
for m, n > N. The contention is now clear.
Next, we observe that
KHM ={0,0, ...,0,1,1,...}
= {1, 1,...} —{1,1......1,0,0,
Now let us show that M is, indeed, a maximal ideal. Suppose
I is an ideal of A, and 2D M properly. Let {a„} e/ and
{an} ф M, i.e., {an} is not a null sequence. Let {£„} be the corre-
sponding Cauchy sequence constructed in (1.3). Then
{M {«Je7,
i.e.,
{1, 1,...} — {1, 1,..., 1,0, 0, ...}£/,
but {1, 1, ..., 1, 0, 0, ...} is a null sequence, and is, therefore,
also in I, so
{1,
which implies that I = A. Hence, M is maximal, and k — A/M
is a field.
We introduce a valuation on h, by defining for a — {an}-\-M e k,
| a | = lim | an |.	(1.4)
We must show first of all that the limit in (1.4) exists, that it is
well defined, and, finally, that is satisfies the axioms for a valuation.
Since
11 I I || |	I,
1. Completion of a Field	29
| an | is a Cauchy sequence of real numbers, and, therefore, has a
limit. Next, if
Ы + M = {bn} + M,
then
so
II &n I I II | I *
and lim | an | = lim | bn |. Finally,
| a | > 0, and — 0 if and only if	lim | an | =0,
i.e., if and only if {a,} e M, or a = M.
If jS — {£„} + M. Then aj8 = {a„M + and
| aj8 | = lim | anbn | = lim | a„ | lim | bn | = | a | | jff |.
« + Д = {.^n +	so
I “ + P | = lim | an + bn | lim | an | + lim | bn | = j a | + | Д |.
Now let a e k. Consider the mapping:
f: k-f-A/M
given by
where {a} designates here the Cauchy sequence a, a, a, ... . The
claim is that/is an isomorphism of k onto
k = {{a} + M\aek}. ’
fab) = {ab} + M = ({a} + M)({b} + M) =f(a)f(b).
f(a + b) = {a + b} + M = ({a} + M) + ({b} + M) ^f(a) +f(b).
If/(a) — 0, then {a}eM, i.e., {a} is a null sequence; whence,
| a | =0, and a — 0. From (1.4),
|/(a) | = lim | a | = | a |,
30 II. Complete Fields and Field of p-Adic Numbers
and f is, indeed, both an isomorphism and an isometry. We will
identify k and K.
Next, we show that k is dense in k. Let a = {an} + M e k, and
let e > 0 be arbitrary. Then | an — am | < e for n, m > N. We
consider the element
= {an , an , an ,...} + M e k,
where n > N:
| Д — a | = lim | an — am | e,
m-wo
so k is dense in k.
Now, we show that k is complete. First, let a^, a2',	, ...
be a Cauchy sequence of k, where
&n	> •••> &n > ••} 4*
Since | a„' | = | an
> ^2 i •••>	, ...
is a Cauchy sequence, and
lim a,[ = a,
where a = {an} + M since
lim I a — a' I = lim lim I am — an I — 0.
П-WO	П-WO 7П-ЭСО
Now, let oq , a2, ..., a„ , ... be an arbitrary Cauchy sequence of
k. Since k is dense in k, we can find a sequence
j ^2 J •••>	’ ”*
of elements of k such that
j a„' — a„ | < 1/и, я = 1, 2,... .
Since
1. Completion of a Field
31
a/, a2',	... is a Cauchy sequence of elements of k, and, by
what has just been shown, lim a/ = a, where a e But then
«j, a2 , an , also converges to a since
I « — a„ | < | a — On' | + |	— v.n
Thus, k is complete.
Suppose kr and kz are two complete fields with valuations
| |г and | 12 such that k is dense in kx and ft , and such that
| and | |2 both extend the given valuation | | on k, i.e.,
| a = | a |2 — | a | for a e k. There exists a sequence
al > fl2 > an > "•
of elements of k such that lim an = <xx where e kr since k
is dense in kt. {«„} is a convergent sequence, and is, there-
fore, certainly a Cauchy sequence. It can, moreover, be viewed
as a Cauchy sequence of k2 since ft Э k and | |2 extends | |.
Since ft is complete lim «„ = a2 , where a2 e k2 . We now define
a mapping,
/:	—> k2 ,
by /(cq) = «2  We contend that f is a congruence and that
f(a) — aforaek.
First, let us observe that the mapping is well defined. If
lim = lim ft = aT
in k1 , and lim an = a2 in k2 , then
I “a — ft |2	| a2 —	+ | an — bn |,
and it follows that lim bn = a2 in ft .
If lim On = and lim bn = ft in kr , then lim (an -f- bn)
=	+ ft in kr. While, if lim an = a2 and lim bn = ft , then
lim (an ft) = ой, + ft in k2 . From which we get
32 II. Complete Fields and Field of p-Adic Numbers
Similarly,
/(“A) =/(«x)/A)>
so/is a homomorphism./is clearly onto ft .
Furthermore, if
lim an = 04 ,	and	lim bn = fa in fa,
and
lim an = a2 ,	and	lim bn — fa in fa ,
then, since
I “1 — & 11 I al an 11 + I an — ft I + I b„ — ft |i ,
we get
I “I ~ Pl 11 < lim I On — bn l>	O-5)
but
I an - bn I 1 an — «1 |i + I «1 - ft 11 + 1 Pl — bn |1.
Hence,
lim I an ~ bn I < I «j - ft |i.
Comparing (1.5) and (1.6), one sees that
lim I - bn I = I ai - ft |i .
A similar argument shows that
lim J an — bn I = I a2 - ft |2.
Thus,
I aI ~ P 11 = I a2 ~~ P2 I 2 »
which shows that / is an isometry, and, therefore, certainly a
one-one mapping, and we have shown that/is a congruence.
(1-6)
(1-7)
(1-8)
2. p-Adic Numbers
33
Finally, for aek, the sequence {«} converges to a in both
and k2 , so f(a) = a.
Summarizing and recalling our identification of k and k, we
have:
Theorem 1.3. Given a field k and a valuation | | on k, there
exists a field ft (unique up to a congruence), called the completion
of k with respect to | such that k is a complete field with respect
to a valuation extending | (, and k is dense in k.
It is clear that ft has the same characteristic as k, and if the given
valuationisnon-archimedian,then so is the extended valuation on k.
Speaking somewhat loosely, we can summarize the completion
process by saying that we associate with each Cauchy sequence
which does not have a limit in k an “ideal” element and to equiva-
lent Cauchy sequences we associate the same “ideal” element.
These elements are then adjoined to k to yield the completion.
We shall denote by | k | the image of k in R under the valuation
mapping | |. We can now establish the following theorem.
Theorem 1.4. If] | is a non-archimedian valuation on k, then
| k | = | k |, where k is the completion of k.
Proof. Let a e If a = 0, | a | — 0. Suppose a 0. Since
k is dense in there exists a Cauchy sequence {an} of elements of
k such that lim an = a. However, | | is non-archimedian, so
I an | = | ex + (an - a) | = max (| a |, j o„ - a |) = | a |
for n sufficiently large by Theorem 2.2 of Chapter I since [ a | =/= 0
and | an — a | can be made arbitrarily small by taking n suffi-
ciently large. Thus,
I an | = I « I
for n sufficiently large, which establishes the theorem.
2. p-Adic Numbers
The completion of the field Q of rational numbers with respect
to a p-adic valuation ! |„ is called the field of p-adic numbers and
3-4 II. Complete Fields and Field of p-Adic Numbers
will be denoted Qv . We shall also use | for the extension of
I Ip Qj> •
Let a &QP , and a 0. We know by Theorem 1.4 above that
I Qv Ip = I Q Ip = U P \Vn I « = 0, ±1, ±2, ±3, ...}
so
I « Ip = \P Ip”’	(2.1)
and, therefore, a/p” = /3 is a unit, i.e., | |„ = 1. We shall denote
by V, the valuation ring of | |„ on Q, by P, the unique maximal
ideal of V, by P, the valuation ring of | |„ on Qv , and by A the
unique maximal ideal of A
Clearly /3 = oc/pn e P. But /? = lim , where ck G Q. Thus
I P ~ ck |p < 1 for A sufficiently large, say, for k N. Hence,
I cn |p = | P + (cN ~ £) |p = max (| p I, , | cN - p |„) = | p .
Therefore,
I CN Ip I P Ip 1’
where cN e Q, and, therefore, cN e V. We now write cN = bn , so
| bn |p = 1, and 1,8 — i„|p<l, whence p + P = bn + A
Let bn — enjdn , where en , dne Z, and where en and dn are prime
to p, which is possible since | bn |p = 1. Thus, there exist integers
x and у such that
xdn 4- yp = 1, or xdn 1 (mod />).
Then
- enx = £n^ ~	= 0 (mod P)
i.e., bn — enx G P, and, hence, bn — enx c A
If we let enx — an , then an G Z, and
P + p = bn + P = an + A
2. p-Adic Numbers
35
Now, | an — /3 |„ < 1, and we have
I anp* - ft* |„ < | p* I, ,	(2.2)
or
a =	= anpn + (fi — an)pn
= anpn + У1 ,
where уг = (fi — an)pn, and | yt |„ < | p |„n by (2.2).
Thus | yj |„ = | P U"1, where m > n, which is the same type
of relation as we started with for a in (2.1). Hence, we treat as
we did a and continue the process. After k steps, we get
a = anpn + Wwl + -• + ‘W-iP’1^"1 + У* ,
where the e Z and | at |„ = 1, or a,, — 0 and where
lyfc|„^lPi;+*.
Since | p \р+к —> 0 as k —>• °°, we have shown:
Theorem 2.1. Any p-adic number ex can be written in the
form
« = X а^’	(2-3)
n
where the a} G Z, and n is such that | a |„ — | p I*,”.
The integer coefficients of (2.3) are only unique modulo p. If
we agree to choose the a, such that 0 sC a3-^ p — 1, then (2.3)
will be called the canonical representation or expansion of a.
We illustrate by an example the canonical expansion. We
shall determine the expansion of f in Qb, and shall adhere
to the notation used in establishing Theorem 2.1. Since
| |6 — | 5 |s° = 1, we see that n — 0. A solution of
8x = 1 (mod 5)
is x = 2, and, since 2  3 s 1 (mod 5), we get that aQ — 1. Now
У1 = (8 ~ 1) = ~ "I >
36 II. Complete Fields and Field of p-Adic Numbers
and | — |5 = | 5 Is1, which indicates that av =£ 0, and
— (5/8 • 5) = — . Again, a solution of
8x == 1 (mod 5)
is x — 2, and 2(—1) s 3 (mod 5). Thus — 3. Next,
y2 = (—i — 3) 5 = (—25/8)5, .
so | y2 |s = | 5 |s3, which indicates that a2 = 0, but a3 0, and,
since
У2/53 = - |,
we see, as above, that aa — 3. Continuing, it is easy to see that
a4 = ae ==•••== 0, while a5 — л7 =	= 3.
The expansion of a. e Qv , say
« =	+ •• + «0 +	+ <W>2 + -
pv p
is frequently abbreviated as follows:
a = a_va_v+1 a0 , ага2 ••• (/>).	(2.4)
For example, we could write the canonical expansion of f in
Qs as
j = 1,30 30 30 -(5),
or, still shorter, as
1,30 -(5),
where the bar designates a periodic repetition.
If the element a e Qp has an expansion (2.3) for which л /> 0,
i.e., if
« = <%, ага2  (p),
2. p-Adic Numbers
37
then a is called a p-adic integer. This will be the case if and only if
I “ Ip = 1P li>” n i-e-> and onty *f a G valuation
ring of | | „ on .
Let a be an arbitrary p-adic integer, a = E^ arf3. We let
sn = Eo 1 аф’. fpn is the principal ideal generated by pn in the
ring V. Then
a = />" (X = Pn&
j—n
where )3 — EJtn a3pJ~ri 6 F, so
a = sn (mod Vpn), n — 1,2,...	(2.5)
where
sn+i sn (mod Vpn),
and each sn e Z. In other words, any p-adic integer satisfies an
infinite system of congruences of the form (2.5).
Recalling our remarks immediately following Theorem 1.3 of
the preceding section section and the subsequent representation
(2.3) for any a of Qv , we see that one could have initially defined
the p-adic numbers as the totality of all formal series of the form
En a,p3, where the a, e Z, and where two such series ^a^p1,
^bjp1 are considered equal if there exists an integer N such that
b = Ь + 0p9
for all j N, where p and p- are the sum of the first j terms of
E ajpj and E bjp3, respectively, and where _j3 e Q and can be
written as fraction with denominator prime to p. Then define
the operations of addition and multiplication of two series' in the
natural fasion. The totality of all such series with respect to the
two operations could then be shown to be a field.
Finally, we shall illustrate the arithmetic operations in Qp using
the notation introduced in (2.4). Before proceeding with specific
38 II. Complete Fields and Field of p-Adic Numbers
examples, we note the following facts which will be useful for the
examples. Since
+ а*+1?<+1 = (ai + />)/•* + («i+i — l)pi+1.
we have
v+1	«0 > ”*	”* =	v-bl ^0 >
at +p ai+1 — 1 ••• . (2.6)
Similarly,
<2—v	, ’ *'	= у * * *	, * * *	p 4" 1	•
(2.7)
Equations (2.6) and (2.7) together with the next observation
will enable us to write all elements obtained in our examples
readily in the canonical expansion. We can write, say
a-, =
where 0 a_v p — 1. Then
< P~v + (* +
Next, we note that
0 = 00 -p(p - 1) - (p - 1), (p - l)(p - 1) - .
Now, if a = S™ Ujpj and /3 = Ь,р\ then one obtains the
expansion of а. Ц- /3 by adding “componentwise,” and to get
a/3, one multiplies the series formally and collects coefficients of
the same powers of p, i.e., the usual Cauchy product, since
one has convergence of the series with |	|„ in place of
and | bjpi |„ in place of Ь?р\
We consider now some specific examples and shall tacitly make
use of the obervations made above.
2. p-Adic Numbers
39
(1)	Addition, (a) In Q7 add the following;
452, 1 3 7 6 1 2
+ 37, 52 1 3 1 52
111 11 1
413,06 1 3 303
(b) In add the following two elements and express the result
in terms of the canonical expansion: | -f- We saw earlier that
inQs
j = 1,303030 •••
while
| = 10,00000 - .
Therefore,
t + | = 11,303030
(2)	Multiplication, (a) Q7 multiply the following:
12,314-1,20 3
12 314
2 4628
3 69 3 12
1112	1
1 4, 0 4 6 4 5 5 1
(b) In Q6 multiply and and express the result in canonical
form. Clearly,
10, 000	1, 3030 ••• = 13, 030303 •••.
(3)	Subtraction, (a) In Q7 subtract the following:
5 6, 3 5 2 4
- 1,2403
5 5, 1 1 2 1
40 II. Complete Fields and Field of p-Adic Numbers
(b) In Q5 subtract the following:
0 5 4, 4 4 4 4 4 4 -
22 1,43021
-13 4, 231422
141,64323244-
= 141, 10423244-
(4)	Division. In Q5 divide 32,13 by 43, 12
43, 12 | 32, 13 = 2,034430034430
32 34
0 3444 -
3234
11044 -
10241
134244-
10241
323244 -
3234
00344 -
In the field R of real numbers one frequently writes a given
number as a decimal expansion. The canonical expansion is the
analog in Qp . Also, in R an element is rational if and only if its
decimal expansion is periodic. The analogous statement holds
for^.
Theorem 2.2. An element a e is rational if and only if its
canonical expansion,
00
X а^'	0 < «, < p,
j==n
where n is such that | a [p = | p \vn, is periodic.
Proof. (1) Suppose the canonical expansion of a is periodic.
2. p-Adic Numbers
41
Then we can write a in the form
a = OnPn + an+iF+1 + "• + an+kPn+k
+ ^«+*+14- b2pn+k+2 +--------h Ь^рп+к+]'
Ц- b1pn+k+i+1 4- ••• 4- bjpn+k+2i
(2-8)
= ?n(«n + On+iP + "• + «n+fc/5*)
4- pn+M(br + b2p+-+ b^-1)
+ рп+к+1+1(Ь1 4- b2p 4- ••• + bfp>-')
+
= pnA +pn+k+1B(\ 4-	4- p2i 4- •••)	(2.9)
where
A = On 4- «n+i? H------F an+kPk,
and
В - К + b2p + ••• 4- b^~k.
Then (2.9) equals
since, in Qv ,
i _ pH	1
1 + pj + P2i + - +	тАз-
as t->oo. However, (2.10) clearly is a rational number, which
completes the first part of the proof.
(2) Assume, conversely, that <xeQ. If we can write a in the
form
р”[Аф — 1) — Bpi+1]
pi _ i
(2.U)
where A, В G Z, and where 0 A < pk+1, О В < pj, then
42 II. Complete Fields and Field of p-Adic Numbers
it follows that a has a periodic canonical expansion, for we can
write (by the canonical expansion or see Appendix)
A — an T an+lP + '' ’ + an+kPk<	0 s2 at < P
В = br + b2p+ • • • + bjpP-1,	0 C bi <p,
and
a = pnA 4- pn+k+1B -—~~pJ ’
which implies that a can be written in the form (2.8) as we saw
in the first part of the proof. Thus, we must just show that a can
be written in the form (2.11). Let | a — \p \Pn, so a]pn =? /3,
where | Д |„ = 1. Hence, we can write /3 = c/d, c, deZ, and
p-X d, p< c. Then there exists an integer j (see Appendix) such
that
pJ = 1 (mod d).
Now
n	- 1)
P d(^-l)’
and since d | pj — 1, we have
p
а = рп^рп --------- (2.12)
p1 — 1
where e G Z. Next, we choose an integer k such that
0 e < />fc+1 if a 0; —p/c+1 sC e < 0 if a < 0.
Since (pk+1, pj — 1) - Appendix) pk+rx uniquely modulo pj — 0 В pi' — 2 if a 5 pk+lB	= 1, we can solve the congruence (see = — e (modp3 — 1) 1. Let В be a solution, where we take 5 0, and 1 В	p]’ — 1 if a < 0. Now, == — e (mod p’ — 1),
3. Some Analysis in Q„
43
and, therefore,
e = A(p - 1) - Bp*+\	(2.13)
where A g Z. It is easy to see that in both cases 0 A < pk+\
Finally, substitution of (2.13) in (2.12) gives
— Рп№Ф — !) — Bpk+l]
a ~	p* — 1	’
where 0 A < pk+1, 0 В < pi, which completes the proof.
3. Some Analysis in Qp
In this section we shall consider the exponential, logarithmic,
and binomial series in . Before doing so, we remind the reader
of some facts already established (II, 1, Theorems 1.1 and 1.2)
in a more general framework; namely, if an tQv{n = 0, 1, 2, ...)
and x e Qv , then E“=o апхП converges if and only if lim = 0.
Also, E” 0 anxn converges if S„=o | anxn converges.
A series of the form E„=o anxn is called a power series. The set of
all x for which E„=o anxn converges is called the domain of
convergence of the power series, and, as in the case of the fields
R or C with with the customary absolute values, one readily
establishes that E„=o апхП converges for | x < r and diverges
for | x > r, where
1
r — =—.........-.
lim v| On |„
The case | x |„ = r is slightly easier to analyze here because of the
non-archimedian character of the valuation; namely, for | x = r,
we have convergence if lim | an |„г" = 0 and divergence if
lim | an \TJrn ф 0.
Other results follow readily as in the case of R or C. In many
cases the proofs can be simplified slightly since | |„ is a non-
archimedian valuation. Thus, one has that if S anxn and S brtxn
converges, then S (a„ + 6n) xn converges and equals £ anxn
± fjbnxn. Similarly, the Cauchy product of '£anxn and £бпяп
converges and equals E anxn • E bnxn.
J
44 II. Complete Fields and Field of p-Adic Numbers
If f(x) = S”=o converges, then lim anxn — 0, i.e.,
lim j anxn j, — 0. D enoting/'(x) as
00
/'(x) = X^""1-
n=0
we have | nanxn 1 С | anxn 1 I? —* 0.
Therefore, /'(x) converges. Moreover,
=	(3.i)
J ' ’	3/-.0	у	'
where у eQv , and у	0. Equation (3.1) follows readily from the
following relations:
CO	00
/(*+y) —fw = X a^x+~ X
n=0	«=0
so
/(x + y) -/(x)
У
xn~jyj—y
(3.2)
But for | у < | x , we have
«и (”) I < I On lp | x |« 1 -> 0,
\y/	Ij)
i.e., the series in (3.2) converges uniformly for all у with
IУ L < i x L • Hence,
lim /(^ +^)
1/-»0	у

xn-jyi-l
co
= ^n^x"-1 =/'(x).
n=l
3. Some Analysis in Qv
45
The other results follow just as simply as this one, and we shall
make use of these results and similar ones in the sequel without
further discussion.
It will be advantageous, for the discussion that follows to
introduce some new notation. We know [see II, 1, Theorem 1.4
and also the beginning of Subsection 2] that if x G Qv , x Ф 0, then
i«i„ =
(33)
where и = 0, ±1, ±2........ We denote by ordpX, called the
ordinal of x at p, the integer n occurring in (3.3). Thus,
1=1* lordsa,
I r Ip >
(3-4)
or | x |„ = cord®“, where c = | p |„, and 0 < c < 1. For x — 0,
we define ordpX = It is clear that
| x |p = 1 о ordp x = 0.
| x |p < 1 о ordpX > 0.
I x |p > 1 ordpX < 0.
(3-5)
We also have that
ordp xy = ordp x + ordp у,
(3-6)
which follows immediately from | xy |₽ = | x |p | у |P . Also, from
I x + У |p < max (I x |p » IУ |p)> one gets
ordp (x + y) min (ordp x, ordp y).
(3-7)
We know [I, (2.12)] that if xn£Q„ and xn -> x, then
| xn |p —> | x |p . In terms of ordinals, this means that if
xn —> x,	then ordpXn —> ordp x.
(3-8)
As a final consideration before discussing the exponential
function, we establish the following useful lemma.
46 II. Complete Fields and Field of p-Adic Numbers
Lemma 3.1. Let n be a positive integer, and let
n = a0 + axp + ••• + й4р‘, О С p — 1
be the canonical representation of n. Then
ordj,(n!) = ^,
where sn = a0 + ax 4- ••• + at.
Proof. For notational convenience, we shall write s0 = 0, Let
1 k n, and suppose
k = 0, 000 •••0 bj>„+1 bt,
is the canonical expansion of k where bv 1, i.e.,
k = bvpv + ^+iP’’+1 + ••' + btp*, sk = b„ + b„+i + ” + bt.
Then
*-l - (P - 1) + (P - 1)? + + (P - ПГ’1
+ (br - 1)r + wr+1 + ••• + btp‘
and
Sfc-i = v(p — 1) + (b„ — 1) -f- b„+t + ••• + bt
= v(p — 1) + sk — 1.
Hence,
_ 4c-l ~ fjt + 1
v p-i '
but clearly v = ord,, k. Therefore,
ord^^^L J^
and, by (3.6), we have
ordP (и!) =	= Пр 11^ ’
which establishes the lemma.
We shall now consider the exponential series E(x) = 2„=0 xnjnl.
3. Some Analysis in Q„
47
First of all, we wish to determine the domain of convergence of
this series. The answer is given in the following theorem.
Theorem 3.1	. The domain of convergence of the expo-
nential series, E(x) — ^n„oxnlnl, is the set of all x for which
ord„x > l/(p — 1).
Proof. By (3.6) and the lemma, we have
ordj’ &)= ” ordp x ~ ord” (m!)
= nordpx-
= n (ordp x — —Ц-) +	.
\	p — 1/ p — 1
Thus, if ord,, x > l/(p — 1), then ordj, (х"/я!)i.e.,
| xn/n! > 0, and the series converges. If, however, ord„x
~ О, then for all n of the form pk, sn — 1, and, for such
n, ordP (х”/я!) •+♦ Thus, the series converges for those and only
those x with ord„ x > l/(p — 1).
In particular, if x e Z and if/» is an odd prime, then for x — kp,
k&Z, ordpX^ 1 > l/(p— 1), and the exponential series
converges for such an x. If however, p — 2 and if x G Z, then x
must be a multiple of 4 in order for x to belong to the domain of
convergence of the exponential series.
It follows immediately from the unconditional convergence of
our series that if x and у belong to the domain of convergence of
the exponential series, then
E(x + y) = E(x) E(y).	(3.9)
We also note some other relevant facts concerning the exponential
series, which will play a role subsequently. If
ord„ x > ——- , x 0,
” p — 1
then	(3.10)
(xn \
—r) > ord„ x for n > 1.
nl / v
48 II. Complete Fields and Field of p-Adic Numbers
The result(3.10)follows immediately from the following relations:
/ л:” \
ord„ — ordy x — n ordc x — ordp n! — ord„ x
fl	s
= („-!) ord, T + j^
П — 1 П	Sn Sn~ 1 > A
" p-1	/>-l+ p-1	p-1
Finally, we note if x belongs to the domain of convergence of the
exponential series, then
ordp (E(x) — 1) = ord„ x.	(3-11)
Equation (3.11) follows from the following considerations:
д*)_1 = л + _+- + -+->
or
E(x) — 1 = lim s„(x),
where sn(x) = x + я2/2! + ”• + xnln\ . Recalling Theorem 2.2
of Chapter I and (3.10) above, we have
ord,, хи(л) = ord„ (x + -jf + '	= ordp x.
Now,
ordj, (E(x) — 1) = ord,, lim sn(x)
— lim ord,, sn(x) [by (3.8)]
-- lim ordp x — ord„ x,
and (3.11) has been proved.
We turn now to a consideration of the logarithmic series.
3. Some Analysis in
49
Theorem 3.2	. The domain of convergence of the logarithmic
series, log(l + x) —	(—1)п-1хп/я, is the set of all x for
which ords x > 0.
Proof. We have
(3-12)
However, it is clear that
pord^n n>
and, therefore,
Inserting this last estimate in (3.12) shows that if ord„x > 0,
then
and the series converges. If ord„ x -C 0, then for those n such
that/» < n, ordP n = 0, and
Hence, the domain of convergence consists precisely of those x for
which ordp x > 0.
It follows readily that the logarithm function satisfies the
functional equation
log (1 + x) (1 + y) = log (1 + x) + log (1 + y),	(3.13)
where ord^x > 0, and ord^y > 0. We also note the following
facts which are analogous to (3.10) and (3.11): if
(3-14)
50 II. Complete Fields and Field of p-Adic Numbers
then
, (_Пп-1лп ,
ord„ I—---------- > ord_я for n > 1,
* \	♦» I	v
and
ord,, (log (1 4- x)) = ordp x.	(3.15)
Once (3.14) has been established, (3.15) follows form it in the
same fashion as (3.11) followed from (3.10). Thus, we just prove
(3.14). Let ordp a; > !/(/> — 1), and x ф 0. Then
ordp
(—I)”-1*”
n
— ordp x = n ordp x — ordp n — ordp x
= («—!) ordp x — ordp n
n — 1
y=Tf-°rd’”

ordpW\
n — 1/
(3.16)
Now, let us suppose that n — paa, where p < a, so ordp n = a.
Then
ordp» _ a	a
n — 1 paa — 1 ' pa — 1
1	a	1
p — 1 ра~г +•••+/> + 1 ""p — 1
and it follows from this and (3.16) that
/(—l)n^1x”\	„
ordp I—--------1 — ordp x > 0.
We are now in a position to show the interrelation between the
exponential and logarithmic functions, namely:
3. Some Analysis in Q„
51
Theorem 3.3	. If ord,, x > l/(/> — 1), then (1) log E(x~) = x,
and (2) E(log (1 + x)) — 1 -f- x.
Proof. (1) is clear for x — 0. We next observe that log E(x)
converges since, by (3.11),
ordp (E(x) — 1) = ordp x > ------r > 0.
P — 1
Finally, since E'(x) — E(x), we have
(logE(x))' = ^® = 1,
so log E(x) = a 4- x, where a is a constant in Qv , and for
x = 0, we get a = 0; whence,
log E(x) = x.
Statement (2) is also clear for x = 0. Next, E(log (1 4- x))
converges since, by (3.15)
ordp (log (1 + x)) = ordp x > j-—.
Now, by the first part of the theorem, we have
log E(Iog (1 4- x)) = log (1 4- x),
which implies (see Exercise 7 at the end of this chapter) that
E(log (1 4- x)) = 1 4- x.
We conclude this section with a brief discussion of the binomial
series.
Theorem 3.4	. Let у eQp be such that ordp у 0 (i.e., ye^>
the valuation ring of | on Qp). The binomial series,
(1 4- x)v — S„=o («) x"> converges for all x with ordp x > 1 /(p -1).
Proof. Since ordp у > 0, and since
= У^У ~ 0	~ 1))
\n/	nl
52 li. Complete Fields and Field of p-Adic Numbers
it is easily seen that
Thus,

but, as we saw in the proof of Theorem 3.1, ord„ (xn/«!) -* 00 if
ord,, x > l/(p — 1), and this completes the proof of the theorem.
From the proof of Theorem 3.4, we have for all у with
ord„y > 0
forord„x > l/(/> — 1). Thus the series converges uniformly in у,
and since each (®) is clearly a continuous function of у, it follows
that (1 + x'Y is a continuous function of y.
Finally, one can show for y^Qj,, ord„y > 0, and
ord„x > I/O — 1),
(1 + x)® — E(y log (1 + x)), and log (1 + x)® = у log (1 + x).
We leave these final considerations to the reader to justify.
4. Newton’s Method in Complete Fields
Let k be a complete field with respect to a non-archimedian
valuation | j; also, let V be the associated valuation ring. We
might think, in view of the applications that will be presented
later, of k as ap-adic field and of | | as | |„ . We want to show
that if certain polynomial equations have an approximate root
in k, then they have a root in k, i.e., we shall extend Newton’s
method to this case. More precisely, we prove:
Theorem 4Л. Let/(x) be a polynomial with coefficients in V,
the valuation ring associated with | | in the complete field k.
4. Newton’s Method in Complete Fields 53
Suppose also that f(x) has leading coefficient 1. If there exists an
a* e k such that
l/fa)l<l and |/'fa)| = l,
then the sequence
“2 = «1
/fa)
/'fa)
“s =
/fa)
/'fa)
(4-1)
converges to a root a e V off(x).
Proof. Observe first of all that eq e V, for if
f(x) = xn + q---------------F a„ ,
where the а{ e V, then
/fa) — ai +	1 + ” + «о •	(4-2)
If | ax | > 1, then the first term of (4.2) would be the dominant
term with respect to the valuation, and we would have | /fa) [ > 1,
a contradiction.
Now, by Taylor’s theorem,
f(x + h) = f(x) 4- hf'(x) + kzg(x, h),
namely,/(x) —	a,x’, where a, e V, and an = 1.
Then
f(x 4- Л) = 2 a^x +
1=0
= /(*) + hf(x) 4- Л2/2(х) + Л3/3(х) + •• + hnfn(x~)
= f(x) + hf(x) + №g(x, Л),
54 II. Complete Fields and Field of p-Adic Numbers
where g(x, h) = f2(x) + kf3(x) + - + hn~2fn(xj.
Thus,
/w=/b-^r)
(43)
but
g (“i>
Now, the coefficients of the /t are all in V, and e V, and
! < 1, so
Hence, from (4.3) and the hypothesis, we obtain
|Ж) I l/(«i) I2-
Again, by Taylor’s theorem
f\x + h) = /'(«) + hF(xt Л),
so

/'(«j) '
but
I -7—7 I < 1, and
1 f (“i) 1
Therefore,
Л“1)
1.
l/'Wl =	= 1-

4. Newton’s Method in Complete Fields 55
Hence, aj satisfies the same two assumptions as , and we may
proceed with the iteration. Thus,
I «2 — «11 = l/(“l) I
I “3 ~ «2 I = 1/(«з) I l/(«l) I®
I “4 — “3 I l/(«l) I*
I «n - “n-i I C l/Ь) I2”"
Finally, set
“ = “1 + («2 — “1) + («з — «2) +
= lim an.
The series converges since its rath term approaches 0. Also, since
it is clear that /(a) = 0. We also note that | a - cq | = lim | an - ax | ,
but
I ocn — a, I < max I a,- —	1 < 1,
so I a — otj I 1, and a e V. The proof is now complete.
We wish to modify the hypothesis of the preceding theorem just
slightly in view of a subsequent application, namely:
Theorem 4.2. Let f(x) be a polynomial with coefficients in V,
the valuation ring associated with the valuation | | in the complete
field k, and suppose the leading coefficient of f{x) is 1. If there
exists an ax e k such that
Ю<1. /'K)^o, |/'(«i)l^i,
and for
then the sequence (4.1) converges to a root a e Fof/(4
56 II. Complete Fields and Field of p-Adlc Numbers
Proof. As in the proof of Theorem 4.1, «j G V. Clearly,
I «2 - «11 = I <4! I-
Again, by Taylor’s theorem,
/(«,)=/(«,-
= /<«.)-	-ft «4)
where fl G V since
Thus,
I I ^1-	(4-5)
Also,
/'(«2) = f («1 -	=	~ Д~у  У>
where у G V, so
and
1	= \/'Ы I I 1 + d1V I =	|.
Hence,
/'(«2)=/'W^	(4.6)
where e is a unit, i.e., | e ] = 1. Now, by (4.4) and (4.6),
J =	_ (/(«1)2//'(«1)2) - ft = Ж)2. s
2	/W №)2-e2 /W
where S e V. Therefore,
\d2 i < I d. p.
4. Newton’s Method in Complete Fields
57
Proceeding inductively, we get
j “n+1 — «П I — | dn I I j
= 141 |/'(«i) I
4 1412”’11/'(«i) I-
Then, as before, we set
® = “i + (“г — ai) + (“з ~ aa) + •’	'
= lim an,
and conclude that the series converges,/(a) = 0, and a G V.
We now want to consider several applications of the previous
theorems to the case of the p-adic fields. First, take k = Q5 and
f(x) = x2 -f- 1. Let aj = 2. Then
/(2) = 5, so |/(2)|6<1,	/'(2) = 4, so |/'(2) |5 = 1.
Thus, by Theorem 4.1, the sequence (4.1) converges to a root a of
Л*У	_
a2 4- 1 = 0,	i.e., V— 1 eQ5.
We note that it suffices for the applications of either theorem
to the case of Qp to try and find an cq e Z, for, by the remarks at
the beginning of Section 2, since oq G F, the valuation ring of
11®on Qv >
cq 4- P = a 4- A
where a e Z, and where P is the unique maximal ideal of P.
We also note that for the above example, we could apply
Theorem 3.4 of the preceding section, namely,
i vr^5.
у = |gQ5 and ord5 -g- = 0; while, ord5 (—5) = 1 > 1/(5 — 1).
Thus
I
58 II. Complete Fields and Field of p-Adic Numbers
As our next application, we want to consider, in general, when
the equation f(x) — x2 a can be solved in , where a e Z,
and p2 -Г a. Suppose, first, that p | a. If x2 — a has a solution
y/a eQP, then
vT«iP = |	(4.7)
since
I a |p = ] VaValP \\Sa\v\Va\v.
Now, from (4.7) and our assumption,
(4.8)
However, as we know (see beginning of Section 2)
\QA> = {|?l/ l.« = 0, ±1, ±2, -},
which contradicts (4.8). Thus, if p [ a, -\/a ф Qp .
Let us now assume that/» 1 a. To apply Theorem 4.1, we must
see if there exists an oq 6 Z such that
l/(«i) I» = I «i2 “ a I® < 1, and l/'(“i) L = I 2ai I® = L
The first condition means that
cq2 = a (modp),
i.e., a is a quadratic residue modulo p. The second condition
means that p -f 2ax . Suppose p 2, and cq2 =s a (mod />). Then
P I “i2 ~ й>
but p -Г a. Therefore, p -f cq and, since p 2, p -f 2cq . Thus, in
this case, if a is a quadratic residue modulo p, then x2 — a has
two solutions in Qv .
If a is a quadratic nonresidue modulo p, then it is easy to see
that x2 — a has no solution in QP , for if a e Qp and if
a2 — a = 0,
4. Newton’s Method in Complete Fields 59
then a e F, and
a + P — b + P,
where e Z. We then have
a + P = a2 + P = b2 + A
which implies that
b2 = a (mod/>).
However, this contradicts the fact that a is a quadratic non-
residue modulo p. Summarizing, we have:
Theorem 4.3. If p is an odd prime, then the equation
x2 — a = 0, where a e Z, and/»2 f a has:
(1) no solution in Qv if p | a.’,
(2) two or no solutions in according as a is a quadratic
residue modulo p or a quadratic nonresidue modulo p if p f a.
Finally, let us consider the case p = 2, where 2 -Г a. Here we
shall apply Theorem 4.2, for it is clear that one cannot find an oq
satisfying the criteria of Theorem 4.1. We, therefore, seek an
oq G Z such that
I /(ai) la “ j ai2 ~~ a la < L and | |г < h
where
, =	= ai2 ~ a
1 f'W 4cq2 •
Since 2 -Г cq2, we see that it suffices to have 8 | cq2 — a so that
both conditions will be satisfied. Thus, we must be able to solve
the congruence
x2 = a	(mod 8)
with an cq such that
cq = 1	(mod 2),
i.e., oq should be odd. But for any odd integer, cq ,
cq2 = 1 (mod 8).
60 II. Complete Fields and Field of p-Adic Numbers
Hence, we see that if e s 1 (mod 8), then we can use any odd
integer cq , and the criteria of Theorem 4.2 will be satisfied.
Conversely, if a is a solution of x2 — a — 0 in Q2 , then a G
and by II, (2.5),
a = b (mod P23)
where be Z. Thus,
a — a2 = b2 (mod 723),
and it follows immediately from this and from the fact that b must
be odd that
a = t2 = 1 (mod 8).
Summarizing, in this case, we have:
Theorem 4.4. The equation x2 — a = 0, where a G Z, 4 < a
has in Q2
(I) no solution if 2 | a\
(2) two or no solutions according as a is or is not congruent 1
modulo 8 if 2 -f a.
It is clear that the field QT1 is topologically distinct from the
field Qg for p q since in Qp the sequence p, p2, ...,pn, ...
converges to 0, while this is certainly not the case in Qg (see also
Exercise 12 at the end of this chapter). With the aid of Theorems
4.3 and 4.4, one can also show that they are algebraically distinct,
namely.
Theorem 4.5. If p and q are distinct primes, then the fields
QP and QB are not ismorphic.
Proof. We consider first the case where p and q are odd
primes and, say, p > q. If there were an isomorphism between
Qp and Qg , then the additive and multiplicative identities of the
two fields would correspond, and, consequently, so would the
elements of the rational subfields. If p is a quadratic residue
modulo q, then, by Theorem 4.3, the equation x2 — p = 0 has
a solution in Qq but no solution in OTJ . However, this contradicts
the assumption that the fields are isomorphic, for if a2 — pinQg,
5. Roots of Unity in Q„
61
then the image of a under the isomorphism would be a solution of
the equation in Qv . Now consider the case where p is a quadratic
nonresidue modulo q. Select a quadratic nonresidue, n, modulo
q such that n < q- n certainly exists since q is odd. Then np is a
quadratic residue modulo q. Hence, again by Theorem 4.3, the
equation
x2 — np = 0
has a solution in Qe , but no solution in since n < q < p, so
p2 < np. Thus, the theorem has been established if p and q are
distinct odd primes.
Now let /э be any odd prime. If 2 is a quadratic residue modulo
p, then, using both Theorems 4.3 and 4.4, we see that the equation
x2 - 2 = 0
has a solution in Qv but no solution in Q2. If 2 is a quadratic
nonresidue modulo p, select an odd quadratic nonresidue, n,
modulo p, which certainly exists, for if m is an even quadratic
nonresidue modulo/), tn Ц- p is an odd one. Now, 2n is a quadratic
residue modulo p. Therefore, the equation
x2 — 2n = 0
has a solution in Q„. However, it has no solution in Q2 since
4 -Г 2», but 2/2».
Since all cases have been considered, the proof of the theorem
has been completed.
5. Roots of Unity in Qp
We shall show that the equation
x*-1 -1=0	(5.1)
has exactly p — 1 distinct roots in Qv . Since Qv is a field, Eq. (5.1)
62 II. Complete Fields and Field of p-Adic Numbers
has at most p — 1 roots in Q„ . Let a represent any of the integers
1, 2, ...,/> — 1. Consider
a = a + [av — a) + (a*2 — a») + •••	2
= lim av".
The series is readily seen to converge since the nth term,
a»”+1 _ avn =	- 1),
and
ap"(p-i) == j (mod/»"+1)	(5-3)
by Euler’s theorem (see Appendix). Thus, the nth term approaches
zero, and the series converges. Also,
ap-i = lim	= 1
by (5.3). Hence, a is a solution of (5.1), i.e., a (p — 1) root of
unity. Letting a assume the different values 1, 2, ..., p — 1, (5.2)
is seen to yield/» — 1 distinct roots of (5.1). Clearly each root
a of (5.1) belongs to F, the valuation ring of | |„ onQ„. Also,
different roots belong to distinct residue classes modulo Д the
unique maximal ideal of F, for if a and j3 are roots of (5.1), say,
a = a -|- (av — a) + (apt — «”) +	,
F = b + (6P - b) + (ft”2 - 6”) + ••• ,
where a b, and 1 a, b «4 p — 1, then
a + P = Ц- P
implies that
a + P — b Ц- A
or
a = b (mod/»),
which is impossible.
Recalling the discussion preceding Theorem 2.1 of this
Exercises
63
chapter, we see now that another canonical type representation is
possible, i.e., any у e Qv can be written uniquely in the form
У = 2
n
where the a, are either 0 or are (/> — 1) roots of unity.
Exercises
1.	Find the canonical representation of the following elements:
(a)|ing3. (b)jinp2. (c)^in£>6.
2.	Perform the indicated operations in .
(a)	123,412 (b)	124,131 (c) (34, 121) - (0, 2103).
+ 421,032	- 321,221 (d) (131, 2) + (2,42).
3.	In Q3, what rational number has the canonical expansion
2,0121 0121
4.	In books on number theory (see e.g. Bibliography) it is shown
that
ordp n\
where [x] denotes the largest integer + x. Use this result to give
an alternate proof that
ordp n\ = -----r.
p — 1
5.	Suppose x and у belong to the domain of convergence of the
exponential series. Show if E(x) = E(y), then x — y.
6.	Show that it is not necessarily true that
ordp (—----------) — ordp x > 0
V / *
if ordp x +--------z-
64 II. Complete Fields and Field of p-Adic Numbers
7.	Show if orcL x > -——- and ord„ у > ——r, and if
P — 1	P — 1
log(l 4- x) = log (1 + y), then x = y.
8.	In Q3 determine the first few terms of the canonical expansion
of log 4.
9.	Prove that the equation x2 + % + « = 0, where a e Z, has
a solution in Q2 if and only if a is even.
10.	Use the result of Exercise 9 to give an alternate proof that
Q2 is not isomorphic to Qp for p^2.
11.	Let к be the completion of k with respect to a non-archi-
median valuation | |. If V is the valuation ring associated with | |
on Ze, let F be the closure of V in к. Show that F = F, the
valuation ring of the extended valuation on к. Similarly, show
that P = P, where P is the unique maximal ideal of V, and P,
the unique maximal ideal of F.
12.	Show that the sequence q, q2, ..., qn, ... has a converging
subsequence in Qp, where q is a prime not equal to p. Does
the sequence itself converge in Qv ?
13.	Consider the sequence of primes
2,3,5,7,11,... in Qv.
Find a convergent subsequence.
14.	Using the notation of Exercise 11, and taking k = Q and
11 — 11 „, show that for any e > 0, there exists a finite set of
elements ar, ..., an depending on e in F such that for any ag F,
there exists an with | a — д* j < e.
15.	Using the notation of Exercise 14, show that F is compact.

CHAPTER III
Valuation Rings, Places, and Valuations
1. Valuation Rings and Places
In this chapter, we want to extend some of the notions already
introduced in Chapter I. In particular, we want to extend the
notion of non-archimedian valuations of rank one, which we had
been dealing with throughout most of the first two chapters. We
shall first consider valuation rings in general—a special case of
which was already considered in Chapter I. Then we shall see
the intimate connection between this concept and those of places
and valuations. Next, we shall consider a general extension
theorem for places. This theorem will be basic in discussing the
possibilities of extending a non-archimedian valuation on a
given field to a finite extension field, which we shall come to in
Chapter V. Finally, we shall give an application of our results
to the concept of integral closure.
Throughout this section, К will denote an arbitrary field.
Definition 1.1. A subring V of К is called a valuation ring
if ae К and if а ф V, then a-1 e V.
It is clear that if К is a field and | | is a rank one valuation on K,
then the valuation ring associated with the valuation in I, Theorem
2.3 is a valuation ring in the sense of Definition 1.1. It is also clear
that an arbitrary field К is itself a valuation ring, which we shall
call the trivial valuation ring.
It follows immediately from the definition that if V is a
valuation ring, 1 G V, where 1 is the identity element of the
multiplicative group of the field K.
Vie consider now the set P of non-units of the valuation ring
V, i.e.,
P = {ae V\ а~г ф V}	(1.1)
Lemma 1.1. If P is defined as in (1.1), then аф P a^1 e V.
65
66
III. Valuation Rings, Places, and Valuations
Proof. Suppose афР. If a G V, then, by the definition of
P, аг1 e V. If а ф F, then, by definition of V, й-1 g V.
Conversely, suppose a-1 G V. If a G V, then а ф P by definition
of P. If a ф V, then clearly аф P. This completes the proof of the
lemma.
The set P considered in Theorem 2.3 of Chapter I certainly
satisfies condition (1.1). There we saw that P is the unique
maximal ideal of V. We can prove the corresponding statement	-
here for the set P of (1.1) with the aid of the lemma. First note:
(1) If a + Ьф P, then either афР or Ьф P.	'
Proof. The statement is clear if either a = 0 or 6=0.
Hence, we may assume that both a and b are not 0. We may also
assume, and do, that a]b G V, for if й/6 ф V, then bja e V, since
V is a valuation ring, and the argument would be analogous. Now,
since a + b ф P, by Lemma 1.1
(й + ft)-1 e V.	«
Therefore,	v
= (1 + g (a + 6)-i e V,
and, again by Lemma 1.1, b ф P.
We may reword (1) as follows:
(Г) If a and b belong to P, then a + b e P.
Next, we have	и
(2) Ifa,6G Fandif дб^Р, thena^Pand6^P.
Proof. Since аЬфР, Lemma 1.1 implies that (й6)-1бК
Therefore,
a-1 = (аЬу'Ъе V,
so аф P. Similarly Ьф P.
We may also reword (2) as follows:
(2') If a, b G V and if either a or b belongs to P, then ab G P.
Statements (Г) and (2') imply that P is an ideal of V. If 7 is an	>
ideal of V and if 7 is not contained in P, then I contains a unit
1. Valuation Rings and Places	67
of F; whence, I = V. Thus, P is the unique maximal ideal of V.
Since P is maximal and since 1 6 V, V/P is a field, called the
residue class field, and P is a prime ideal.
Again, we denote by V a valuation ring of K, and we let U
denote the units of V, i.e.,
U = {a G V | a-1 e V}.
It is clear that U is a multiplicative group, which will be called the
group of units of V.
We now claim that the field К decomposes into the following
disjoint union:
A = P v G v (P - {0})-1,	(1.2)
where P — {0} denotes the difference set, and (P — {0})-1 denotes
the set of elements inverse to those of P — {0}. Since P U — V,
in order to prove (1.2), all one must show is that К — V
= (P — {0})-1. Thus, suppose that a e К — V, so а ф V, and,
therefore, a"1 e V, and a-1 e P, so a e (P — {0})-1. Conversely,
if a e (P — {0})”1, then a-1 G P. Hence, a$V, and, therefore,
a e К - V.
It is easy to see from (1.2) that if and V2 are two valuation
rings of К with non-units Px and P2 and units Ux and U2 ,
respectively, then
Vx C V2 о Л Э P2 о Ut C U2 .
The next concept that we want to introduce is that of a place.
Although a place is intimately related with a non-archimedian
valuation, we did not point this out in the introductory chapters,
but now we shall proceed to consider the relationship. Actually,
in a sense, the reader is acquainted with places, as we shall point
out in some of the illustrative examples which will follow the
definitions.
Definition 1.2. Let К and F be arbitrary fields. A map
ф :	{oo} is called a place if:
(1) <£-1(P) = V is a ring;
68 III. Valuation Rings, Places, and Valuations
(2) the restriction of </> to V, Ф | v, is a nontrivial homo-
morphism;
(3) if = oo, then = 0.
We shall give some examples of places.
(a)	Let К = F(x), the field of rational functions of a trans-
cendental element x over F with coefficients in the field К Write
each element of F(x) in reduced form. If we substitue x = a&F
in each element, we obtain a well-defined mapping
К =F(x)->Fv{oo}
under the specification that if, after the substitution, 0 appears in
the denominator then that element is mapped into °°. It is a
simple matter to verify that this mapping is, indeed, a place.
Thus, one might view a place intuitively as a substitution mapping.
(b)	Let К — Q, and let p be a fixed prime. Take F —
where (p) is the principal ideal generated by p in Z. Now, write
any Д 6 Q in the form fl — a/b, where not both a and b are
divisible by p. Consider the following map
defined as follows: replace a and b by the residue classes to which
they belong modulo p, and, should the denominator be the 0
residue class, map the element into 00. Again, it is readily seen that
we have a place. Actually, the first example follows the same
procedure as this one, for, if f{x) еТфс], then, by the division
algorithm
/(«) = {x — a) q(x~)	and f(x) == f(a) mod (x — a),
i.e., in this example, we considered residue classes modulo
(x — a) in the ringF[x],
Now, we wish to draw some consequences from the definition
of a place. The first observation is that if <f> is a place, where
<h : К F и {<»}, then
V = <^-1(F) is valuation ring, called the valuation
ring associated with the given place. The proof is
1. Valuation Rings and Places
69
immediate, for if афУ, then </>(a) = and, (1.3)
therefore, ^(a-1) — 0; hence, a-1 e V, and V is a
valuation ring.
Let us consider the non-units, P, of this valuation ring. By
(1.2), we know that P consists of the 0 element and the inverses
of those elements which do not belong to V. Thus, by definition
of the place ф,
m = o.	(1-4)
Suppose a G V, and <£(a) = 0. If a-16 V, then
ф(Г) = ф^аа-1) = ф(а)ф(а'1) = 0,
which implies that ф(У) = 0, but this contradicts the fact that
ф is nontrivial. This together with (1.4) shows that the kernel of ф
on V is precisely P.
We have seen, up to this point, that, with a given place, there
is an associated valuation ring, V. Now we shall show that the
converse is true, namely, given a valuation ring of a field К a
place can be associated with it. Denote by P the maximal ideal of
non-units of F, and define
(0° if	a i V
+ P if	Av	™
Thus, on V, ф is defined to be the canonical homomorphism onto
the quotient ring V/P, which is a field F. Clearly, ф satisfies
condition (1) of Definition 1.2, and, on V, ф is a nontrivial
homomorphism since 1 ф P, so condition (2) is satisfied. Finally
if а ф V, then a G (P — {0})-1, so a-1 e P. Hence, condition (3) is
satisfied. The valuation ring associated with the place ф is the
given valuation ring V.
The place ф so constructed will be called the place associated
with the given valuation ring.
We return now to the situation where a place ф: К-+F<j {°°} is
given and denote by V, the associated valuation ring, so
Ф\у- V^F
70 III. Valuation Rings, Places, and Valuations
is a homomorphism with kernel P, as has been noted. Hence, the
map </> | V decomposes into the product of 3 maps: </> \v = if к',
where
V-^ VIP-^^tVj-^F,
and к is the canonical homomorphism, f is an isomorphism, and i
is the injection map. If we extend к to К — V by taking к(а) = °°
for all a&K — V, then this would be the place (1.5) associated
with the valuation ring К We can, therefore, state the following:
There exists a one-to-one correspondence between places and
valuation rings, where by this we mean the following: a given
place determines an associated valuation ring, which, in turn,
determines an associated place, which is equal to the given place
up to an isomorphism.
Two places are called equivalent if they have the same associated
valuation ring. Thus, we can reword the above result by saying
that there is a one-to-one correspondence between places and
valuation rings up to an equivalency.
1.	Valuations
We shall, in this section, consider a more general notion of
valuations than we did in Chapter I, and we shall then interrelate
this concept to those of places and valuation rings. Before doing
this, however, we must define what is meant by an ordered group.
Definition 2.1. Let G be a multiplicative group. The group G
is said to be an ordered group if G contains a normal subsemigroup
S such that
G = Su{l}uS-\	(2.1)
where the union is a disjoint one, and where S-1 denotes the set
of all inverses of elements of S.
Now, let G be an ordered group and define
a < b if and only if ab~x G 5, where a, b G G. (2.2)
In particular, a < 1 if and only if a G .S'. We shall show that
2. Valuations
71
< satisfies the customary order relationships. We note first that
a < b will also be denoted at times by b > a.
(1)	a < b if and only if b^a e S.
Proof. Ь~га = 6-1(a6-1) b.
Therefore, if air1 G S, then Ь~га g S since 5 is normal. Similarly,
since
ab-1 = 6(6-1a) 6-1,
if b-1 ae S, then ab-1 G S, which completes the proof.
(2)	a < b, or a = b, or b < a.
Proof. From the decomposition (2.1) of G, we have that
either a6-1e S, and then a < b, or a6-1G{l}, and then a = b,
or ab-1 G S~r, and then ba-1 G S, and b < a.
If a < b or a — b, we write a b, or b a.
(3)	If a < b, and c e G, then ac < be, and ca < cb.
Proof. Since a < b, ab-1 G S, so
аагЧг1 G S, i.e., (ac) (6c)-1 G S,
which means that ac < be. Similarly, one shows that ca < cb.
(4)	If a < b, and b < c, then a < c.
Proof. Since a < b, ab-1 G 5. Also, since b < c, 6c-1 6 5.
Therefore
ac-1 = (ab-1) (6c-1) G S,
for S is a semigroup, so a < c.
(5)	If a < 6, then 6-1 < a-1.
Proof. By property (3), a < 6 implies that 1 < a-16, and this
implies, again by (3); that 6-1 < a-1.
(6)	If a < 6, and c < d, then ac < bd.
Proof. From a < 6, we get, using (3), that ac < 6c. Simi-
larly, 6c < bd, and then, by (4), ac < bd.
In an ordered group G, it is possible that an additive operation
72
III. Valuation Rings, Places, and Valuations
(+) is also defined. However, it is always possible to introduce in
G an additive operation by defining
a + b — max (a., b),	(2.3)
which is well defined by property (2). Also, the operation + is
clearly an associative operation. It can also be seen that the
distributive laws are satisfied. For example, suppose a b, then
ac C be, and max (ac, be) — be. However, max (a, b)e == be.
Therefore ac + be — (a + b) c.
Next, if G is an ordered group, we shall adjoin a zero element,
i.e., form G и {0}, in such away that a • 0 = 0 • a — 0 for all
a e G, and such that 0 < a for all a e G. Then, by (23),
a-0 = 04-a = «,
for any a E G.
With these notions, we can now give the definition of a
valuation.
Definition 2.2. A valuation of a field К is a map | |:
К —>• G и {0}, where G is an ordered group provided
(1)	for a e K, | a | = 0 if and only	if	a	=	0;
(2)	for a, b e K, | ab | = | a | | b |;
(3)	| a + b | < | a | + | b |.
It follows immediately from (2) of the definition	that | 1 | = 1.
We also have that | — 1 | = 1, for, again by (2),
I -i I3 = 111 = i;
however, in an ordered group, G, there exists no element except
the identity with finite order, for if a > 1, then an > 1 for any
positive integer n.
We now suppose that the additive operation in G is based on
the max, i.e., it is given by (2.3); whence, condition (3) of the
definition becomes
| a + b j C max (| a | b |),
and the valuation is this case will be called non-archimedian.
2. Valuations
73
Note that I, Theorem 2.2 also holds for our more general
non-archimedian valuations. The proof is the same.
Next, we shall show that to a given nonarchimedian valuation
there is an associated valuation ring. Let
V = {a e К | | a | < 1}.
(2-4)
V is certainly a ring, for if а, b e V, then | a | C 1 and | b |	1, so
|	| = | a | |6| < 1,
(2.5)
and
| a — b | C max (| a |, | b |) C 1.
(2-6)
Equations (2.5) and (2.6) imply that И is a ring. Next, let us show
that V is a valuation ring. Thus, suppose that а ф V. Then
| a | > 1; therefore,
| a-i | = | a |-i < 1.
Hence, a-1 e V, and V is, indeed, a valuation ring.
The non-units, P, of V are. those a e V such that а1 ф V, or,
in other words, those a&K such that | a | C 1 and | a |-1 > 1,
so
P = {a e К | | a | < 1}.	(2.7)
Consequently, the units, U, of V are
U = {aeK\ ] a | = 1}.
(2-8)
Now, let us analyze a given valuation mapping | | more care-
fully. Denote by K* the set К — {0}. Condition (2) of Definition
2.2 states that | | restricted to the multiplicative group K* is a
homomorphism into the group G. If we denote the image group
by | K* | and observe, by (2.8), that the kernel of this homo-
morphism is precisely U, then we obtain
K*)U ~ I K* |.
74
III. Valuation Rings, Places, and Valuations
Thus we have that | | on K* factors into the product of the
following sequence of mappings:
K*/U-^\K*\++G
(2-9)
where к is the canonical map, f is an isomorphism, and i is the
injection map.
Suppose we now reverse the procedure, namely, suppose a
valuation ring, V, of a field К is given. Let P be the unique
maximal ideal of non-units of И, and let U be the group of units
of V. To the given valuation ring V, we wish to obtain an
associated valuation. From the previous discussion, in particular
from (2.9), the following map suggests itself:
\aU	for	aeK*
|a|=l0	for	a = 0.	<2Л0)
Thus, we take G = A-*/ U and the additive operation in G based
on the max. We must therefore, show first of all that G is an
ordered group. To do this, we must exhibit a subsemigroup, S,
with the required properties. Define 5 as follows: aU e S if and
only if a&P* — P — {0}. Clearly S is a semigroup, and it is
certainly normal since К is a field. Also, by (1.2) of this chapter,
К* — P* u U и (jP*)-1 (disjoint).
Therefore, we get mod U that
G = K*)U = 5 u {1} w S-1 (disjoint)
so G — K*l U is an ordered group.
Now, we check that | a | = aU satisfies the conditions of
Definition 2.2. The first two conditions follow immediately from
the definition of | a |. To verify the third condition, we must show
that | a b |	| a |	| b |. However, this is equivalent to the
following:
(2.П)
2. Valuations
75
For, say, | a/b | < 1, then, if (2.11) is satisfied,
|1+| <1 + If I,
I о	i b I
which implies that | a + b | < | a | + | b |. However, condition
(3) certainly implies (2.11). Therefore, in our case, in which the
addition is based on the max, we must show that | a | C 1
implies that | 1 + a |	1. Thus, suppose that a U = | a |	1.
Then a eV, and, therefore, 1 ae V, i.e., ( 1 + a | < 1.
Hence, | a | — a U is a valuation and, moreover, its associated
valuation ring is the given one V.
Summarizing, we have that a given valuation determines an
associated valuation ring, which, in turn, determines an associated
valuation, which is equal to the given valuation up to an ismor-
phism. If we define two valuations as equivalent if they have
the same associated valuation ring (this, incidentally, is consistent
with our use of the word equivalent for rank 1 valuations in
Chapter I), then we can say that there is a one-to-one corre-
spondence between non-archimedian valuations and valuation
rings up to an equivalency. Since we have already seen that there
is a one-tp-one correspondence between places and valuation
rings up to an equivalency, we now have the complete inter-
connection of the three concepts, and, by identifying equivalent
places and equivalent valuations, we have that there is a one-to-
one correspondence between places, valuation rings, and non-
archimedian valuations. In the sequel, we shall not distinguish
equivalent places or equivalent valuations.
Let us pause for a moment and consider a specific example. We
take К = C(z). The field of all rational functions of a single
complex variable. We know that a place is obtained by substituting
z0 e C for z. The associated valuation ring of this place is
 -
where f(z), g(z) e C[ar]. The maximal ideal of non-units is
P= |^o)^ O,/(*o) = 0[,
lt> III. Valuation Rings, Places, and Valuations
and, finally, the group of units is
u=
The valuation associated with this valuation ring is
| h(z) | = h(z) U,
where h(z) e C(z). Hence
I h(z) I = (z — *o)n u,
and я is called the order of the zero of h(z) at z0 if я > 0, and
—я the order of the pole of h(z) at z0 if n < 0.
The ordered group G here is just the infinite cyclic group
generated by (z — z0) U, and is, therefore, isomorphic to Z.
Finally, we have
| h(z} | < 1 (z — z0)n e P* n > 0.
We see that the given place merely tells us if a function has a
pole or zero at a given point z0 , while the valuation gives us a
refinement of this statement by telling us about the order of the
pole or zero.
3. Valuations of General Rank
We want to consider specifically how the non-archimedian
valuations of rank 1, considered in Chapters I and II, fit into the
general scheme of things discussed in the previous section of this
chapter. In order to do this, we must introduce a general notion
of rank of a non-archimedian valuation, and show that for the
non-archimedian valuations of rank 1 the general value group can
always be replaced by an additive subgroup of the field R of real
numbers which is order-isomorphic to it.
In order to carry out this program, we must introduce some
preliminary notions. Let G be an ordered group as defined in
Definition 2.1 of the previous section. We shall assume through-
out that G 1.
3. Valuations of General Rank
77
Definition 3.1. A subgroup H of G, where G is an ordered
group, is called isolated if Ье H, and b"1 C a C b, where a 6 G,
imply a e H.
Clearly, the whole group G and the identity subgroup, {1},
are isolated subgroups of G. It is entirely possible that these are
the only isolated subgroups of G. This occurs, for example, when
G is infinite cyclic.
We shall now prove the following fundamental fact.
Theorem 3.1	. If Hr and H2 are two isolated subgroups of the
ordered group G, then either Hr C H2 or H2C H1.
Proof. Let Si — {ae Нг \ a > 1}, and S2 = {a E H2 \ a > 1}.
We first note that, if H2 , S2 , for suppose 5^ = S2 .
Let a E Hy. If a > 1, then a E Sx = S2 C H2 . If a < 1, then
a~r > 1, which implies a-1 e Sx = S2 C H2, and, therefore,
a e H2 . Thus Hi C H2. Similarly, H2 C Hx , and H± = H2t
which contradicts the assumption. Thus, if Нг Ф H2 ,	=#= S2 ,
and we may find, say, an element a E Hr , a > 1, and а ф H2 .
Let b E S2. Now, we must have b < a, for, otherwise, b~* a < b,
which would imply that a E H2 since b E H2 and since H2 is
isolated. Thus b < a, but b > 1 > a-1. Hence,
a-1 < b < a.
where a belongs to the isolated subgroup Hr . Thus, bE Hr ,
and we have shown that S2 С Нг , but this clearly implies that
H2 С Нг .
Theorem 3.1 shows that the set of all isolated subgroups of an
ordered group G form a totally ordered set with respect to the
order relation given by set inclusion.
Definition 3.2. Let G be an ordered group. The order type
of the set of all isolated subgroups of G distinct from G is called
the rank of G.
Definition 3.3. If | | is a non-archimedian valuation of a field
К into G и {0}, where G is an ordered group, then the rank of
| | is the rank of G.
78
III. Valuation Rings, Places, and Valuations
Definition 3.4. Two ordered groups G and G' are called
order-isomorphic if there exists an isomorphism: a —> a! mapping
G onto G' such that a < b => a' < b', where a, b e G, a', b' e G'.
As our last definition of this section, we have:
Definition 3.5. An ordered group G is called archimedian if
for every a, be G with b > 1, there exists an integer n such that
bn > a.
• Let us consider some relationships between, and consequences
of, these definitions.
Theorem 3.2	. An ordered group G is of rank 1 if and only if
it is archimedian.
Proof. (1) We suppose first that the rank of G is one. If G is
not archimedian, then there exist elements a, be G such that
b > a > 1, and an > b does not hold for any positive integer n.
Let
5 = {c e G | 1 < c, and c < an for some n}.
Clearly, Ь ф S. If сг and c2 belong to S, say, 1 < ct < an and
1 < £a < then
1 < CjCz < an+m,
and cxc2 e S. Hence, 5 is a subsemigroup. Also, if 1 < d < c,
where c e S, then d e 5. Now let H = [S], be the subgroup of G
generated by S. Since S is a semigroup, H is the set of all elements
of the form
•••
where the e S, and the e, — ± 1. We claim that H is an isolated
subgroup of G distinct from {1} and G. Once this has been shown,
the first part of the theorem will have been established, for this
would contradict the fact that the rank of G is 1, and, hence, the
assumption that G is not archimedian is false.
Now, clearly, H {1} since aeH. Next, H G, for Ьф H,
namely, if
b = c/ic/s ••• c/«	(3.1)
3. Valuations of General Rank
79
where the ct e S, and the et- = ± 1, then for those ct with €,= — 1,
Cif< < 1, and for those Cj with e> = 1,	< an* for some ttj .
Hence, (3.1) implies that
* b < a ,
where the summation is over those j with в,- — 1 in (3.1), but this
means that b e S, which is a contradiction. Finally, H is isolated
for suppose e 6 H, and e~r d e, where deG. Then, if
e =	 • • сп(п, we have
«
•••	< d <- q£i ••• cn£»,	(3.2)
but, as in the argument above,
d cxfl ••• cn£"
implies that d < am, for some integer m. If d > 1, then de S С H.
If d < 1, then d~r > 1, and, by (3.2),
c/1 ... c^n d-i
whence, as before, d~* e S С H, and deH. This completes the
proof of the first part of the theorem.
(2) Conversely, suppose that G is archimedian. Let H be an
isolated subgroup of G where	Choose beH, where
b > 1. Let a be an arbitrary element of G, and say, « > 1.
Then there exists an integer n such that bn > a; whence,
b~n < a < bn,
and a e H. If a < 1, then a-1 > 1, and, by the preceding result,
a'1 e H, so aeH. Thus H = G, and the rank of G is 1. The
theorem has now been established.
The next consequence of the definitions that we shall establish
is the following:
Theorem 3.3. An archimedian ordered group G is abeliaq.
Proof. If we can show that for any two elements a, beG,
where a > 1 and b > 1, ab — ba, then we are clearly done.
80
III. Valuation Kings, Places, and Valuations
Suppose, first that there exists a smallest element ceG such that
c > 1. If a > 1, then, by definition of c, c a. Also, since G is
archimedian, there exists a positive integer n such that
cn a
cn+\
Thus, 1 < acn < c, which implies that ac~n = 1, or a = c”.
Thus, if a > 1, a — cn for some positive integer n. If a < 1,
then a-1 > 1, and a-1 = cm for some positive integer m, and
a — c~m. Therefore, in this case, (^= [c], the infinite cyclic
group generated by c, and G is, of course, abelian.
Next, we suppose that there exists no smallest element of G
greater than 1. We shall first show that if c is any element of G
such that c > 1, there exists an element deG, where 1 < d < c,
and d2 c. Since c > 1, and since G contains no smallest
element greater than 1, there exists an element её G such that
1 < e < c.
(33)
There are two possibilities: either e2 c, or e2 > c. In the first
case, we take d = e as the desired element. Let us consider now
the second case.
e2 > c
(3-4)
and, therefore, e > ce~\ Thus, 1 > and
c > ce~lce~r = (ce-1)2.	(3.5)
Take d — ce~\ Then, by (3.5), we have c > t/2, and, using (3.3)
and (3-4), we have
1 < d = ce~x <e <c,
and d — ce"1 is the desired element.
Now, suppose that a, b G G, where a > 1 and b > 1. Suppose
ab Ф ba. There is no loss of generality in assuming
abtya)-1 — с > I.
3. Valuations of General Rank
81
Then, by the preceding result together with the fact that G
contains no smallest element greater than 1, we can find a de G,
where
1 < d < c, cP c and d < a, d < b.
Since G is archimedian, there exist positive integers m and n such
that
dm < a < dm+1
and
</"<&< </n+1.
Therefore,
dm+n ab -<C <7m+"+2> and
^-(m+n+2) < (bd)~x 3^ jf-On+n).
Thus
d-2 < (ab) (ba)-1 < d2,
or d~2 <_ c <_ d2 c,
a contradiction. This completes the proof of the theorem.
Finally, we establish the following theorem, which will show
how the valuations of rank one considered in Chapters I and II fit
into the general concepts of this chapter.
Theorem 3.4. An ordered group G of rank 1 is order-
isomorphic to a subgroup of the additive group R of real numbers.
Proof. By Theorems 3.2 and 3.3, we know that G is abelian.
If G contains a smallest element c such that c > 1, then, as seen
in the first part of the proof of Theorem 3.3, G — [c], and,
therefore, G can be mapped in an order-ismorphic fashion onto Z,
the additive group of integers.
Next, suppose that there exists no smallest element of G greater
than 1. First, we map the identity, 1, of G into the real number 0.
Next, let a be an arbitrary element of G with a > 1. We map a
into the real number 1. Now, let b c G; to b we associate a real
number b' defined as follows: for m]n e Q, with n > 0, m\n eL(b),
the lower class of a Dedekind cut b', if am sC bn, and m/» e b\b),
the upper class of a Dedekind cut b' if am > bn. Clearly m/n eL(b)
or mjn e U(b). Since G is archimedian, b~n < a < bn for some
82
III. Valuation Rings, Places, and Valuations
integer n. If n is positive l/я eL(i); also, there exists an m such
that am > bn, so m/n e U(b). Thus L(b) ф в and U(b) 0. A
similar argument establishes the same result if n is negative.
If m/n G U(b), and if s/t m/n, where n, t > 0, then ns mt.
Thus, since am > bn, we have an‘ amt > 6ni; whence, as >
and s/t e U(b). It now follows that L(b) and U(b) are indeed, the
lower and upper classes of a Dedekind cut, b', in R.
Suppose that b, ceG and b > c. Then bc~r > 1, and, by the
archimedian property of G, there exists a positive integer n such
that
(3.6)
Let m be the smallest integer such that > cn. For this m, we
must have
(3-7)
since, otherwise, am > bn, which implies, by (3.6), that am > acn,
and then am-1 > cn. This, however, contradicts the choice of m.
Hence, (3.7) holds, and this implies that mjneL(b), and
m/ne U(c)\ whence b' > c', and our correspondence is one-one
and preserves the ordering.
If Wi/nj sL(b) and m2/n2 еОД, then we can write m1jn1 — mln,
and m2/n2 — m'/n. We have
am <- bn, and am' <- cn.
Thus, am+m' (bc)n, so
Therefore, we have shown that
L(b) +L(c) СЦЬс).
Similarly,
U(b) + U(c) C U(bc),
and from these relations it follows immediately that (be) r=b'+c',
and the proof of the theorem has been completed.
4. The Extension Theorem
83
Thus in considering valuations of rank 1, we may always
assume that the ordered group G is a subgroup of the additive
group of real numbers with the customary ordering. These were
the type valuations considered in Chapters I and II, and the
reason for the rank one terminology initially adopted there is now
clear.
4. The Extension Theorem
We shall now establish the fundamental extension theorem for
places, which will play a basic role in the subsequent discussion
on the extension of non-archimedian valuations. In the hypothesis
of the theorem, it will be assumed that a certain field, F, is
algebraically closed. This is no real restriction since, in general,
one could always inject F into its algebraic closure. We now
state and prove the theorem.
Theorem 4.1. Let К be a field and A a subring of K. Let F
be an algebraically closed field, and
f-.A-+F,
a nontrivial homomorphism. Then there exists a placed of К such
that <^> \A — f, where ф \A is the restriction of ф to A.
Proof. Let
S =	|/(Z>) ^0}.
It is clear that 5 is a semigroup and that S 0 since f is non-
trivial. We form the quotient ring
A' —	| a e A, b e S]L
A' is a ring with identity which contains A as a subring, and in
A' all elements of 5 have inverses. We shall now extend f to a
mapping/' defined on A'. This is done as follows: for a/beA'
define	,
л_/(д)
7 W fib)’
84 Hi. Valuation Rings, Places, and Valuations
We first observe that the map f is well defined, for if	= az/b2,
where ax , aze A, and br ,bse S, then агЬ2 = а2Ьг , and since/is a
homomorphism
However, /(^i) Ф 0 and f(b2)	0. Thus, dividing by /(b^f^)
we get
f(b2) ’
which shows that /' is well defined. It follows immediately from
(4.1) that/' is a homomorphism since/ is. Also,/' \A = f, for if
a e A, then a = abjb, where be S, and
/(aft) f(a)f(b)
J \b) f(b) f(b)
This is the first type of extension, but it is possible that it
yields no extension at all. This will be the case when the ring A
is its own quotient ring, i.e., when a G A and f(a) 0 implies that
a-1 g A. When this occurs, we shall show that if we select any
a eK, f can be extended to either the ring Л[а] or to the ring
Л[а-1], and this is the second type of extension that we shall
consider.
Thus, assume that A is its own quotient ring. We first note that
/(Л) —Fo is a subfield of F, for let coeFo and cQ ф 0. Then
e0 = /(а) Ф 0, where a e A. Hence, a-1 G A, and
1 = /(аа-1) =/(a)/(a-i) = c0/(a-i);
Therefore, c0 has an inverse in Fo , andFe is a field.
We shall denote by a, the image of aeA under the homo-
morphism /. We extend / to the polynomial ring A[x] in a
transcendental element x in the obvious fashion by just applying
/ to the coefficients of a polynomial. The image of P(x)e A[x]
will be denoted by F(x). The image of A[x] under this extended
homomorphism is just F0[jc], andP0[x] is a principal ideal domain
4. The Extension Theorem
85
sinceFo is afield. Now we attempt to extend/to a homomorphism
g on Л[а], where a. e K, by defining
^(P(a)) = P(0),	(4.2)
where /3 is any element ofF. If£ is well defined, then it is certainly
a homomorphism which extends /. Thus, we must just see if g is
well defined. This amounts to showing that Р(а) = 0 implies
= 0. Let
7 = {Р(х)еЛ[х]|Р(а) = 0}.
I is clearly the kernel of the substitution map:
Л[х] Л[а],
and is, therefore, an ideal of Л[х]. Hence, to check if g is well
defined, we must determine whether the image I in F0[x] of I is
such that x = /? is a zero of it, i.e., whether P(/3) — 0 for all
P(x) e I. If this is the case, then g is clearly well defined. Since I
is an ideal ofF0[x], I is a principal ideal, and we have
i = ow - ад.
for some Q(x) eF0[x]. Thus Д must be selected so that Q(fl) = 0,
and, since F is algebraically closed, such а Д can be chosen
provided Q(x) is not a nonzero constant polynomial. Hence, if
Q(x) is not a nonzero constant polynomial, we have an extension
of/to£ given by (4.2) on Л[а]. However, it is possible that Q(x)
is a nonzero constant (should Q{x) — 0, then it is clear that any
Д eF may be chosen), in which case our construction fails. Thus,
we must consider this case in more detail.
Clearly, we may assume that Q(x) — 1. Then there is a
Q(x) = 1 + a0 + a->x +-----h a<x‘,
where a, — 0, i — 0,1,..., t, and where
1 -f- Ч- «1“ 4" *" 4"	— 0.
86 III. Valuation Rings, Places, and Valuations
Thus, if a satisfies such an equation, our construction does not go
through. However, we shall now show that if this situation
prevails, then we can extend f to Я [a-1]. For suppose the con-
struction did not work for Л[а] and for Л[а-1]. Then we would
have
1 + a0 + ai“ + " +	(4-3)
and
1 + ad + ad - +  • • + as' — = 0,	(4.4)
where the a{ and a/ belong to A, and where ul = a/=0,
i =0,..., t, j = 0,..., s.
We may also assume that t and s are the minimal degrees of
such equations satisfied by a. and 1 [a, and we may also assume that
s <t since the argument will be symmetric in r and t. Observe,
first, that both t and s are greater than or equal 1, for if, e.g.,
t — 0, then
1 + ao — 0,
and this implies that
1 + ao = 0,
or T = 0, since Oq = 0. However, this is a contradiction since/is
nontrivial. Next, from (4.4), one gets
The coefficients of (4.5) belong to A since 1 + ad — 1	0, and
A is its own quotient ring. Thus, (4.5) can be written as
«s = a'd + a”a +---------F a''-ia’-1>	(4.6)
where the a'd e A, and where a'd = 0, i = 0, ..., r — 1.
Since r t, we can write
a* — a’(«t-*),
4. The Extension Theorem
87
where t — s 0. Now, using (4,6), we can lower the degree of
a in (4.3), i.e., we can write (4.3) as
1 a0 +	+ ‘" + flt(at-e) — 0>
or
1 + «0 + ara. +	+ ^«‘-’(a" + a”a. -j- ••• + л'21«’~1) = 0,
(4-7)
and the highest power of a in (4.7) is t — 1, which contradicts the
minimality of t. Hence, we have finally shown that if we can not
extend f to A [a], we can extend it to Л[а-1].
We are now in a position to apply Zorn’s lemma. Let E denote
the set of all extensions of f to larger rings. In other words,
£ e E if g is a homomorphism defined on a ring В Э A, and
g \A = f- If^i, g<i& E, we define^ > gt, ifgz is an extension of
gi. This clearly defines a partial ordering of E. Let {ga} he any
totally ordered subset of E. The set of rings, {Ла}, on which the
ga are defined are; therefore, totally ordered by ordinary set
inclusion. Form the union of these rings, which is a ring in this
case and define g on this union as follows: if a G Ua Aa, then a
belongs to one of the rings, say Aa , and define
g(a) = £«(«)•
g is well defined since the set {ga} is totally ordered, and g is
clearly a homomorphism. It is also clear that g > ga for all a, and
ge E. Thus, the set E is inductively ordered, and, therefore, has a
maximal element. Let h be such a maximal element, and
h: V^F.
Since h can not be extended any further, we know, by the pre-
ceding discussion, that
(1) V is its own quotient ring by elements with nonzero images,
i.e., if а 6 V, and if h(a) ф 0, then a-1 e F;
(2) if а V, then we know that h cannot be extended to1 F[a]
since A is a maximal element of E. However, this means, as we
88
III. Valuation Rings, Places, and Valuations
previously saw, that h can then be extended to И[а-1], but, since
h is maximal, we must have a-1 6 V. In other words, if a V,
then ar1 e V, or V is a valuation ring.
Finally we show that the place, ф, associated with this valuation
ring is equal to h up to an isomorphism. This follows from (1),
for by (1), if a e V and if h(a) 0, then a-1 e V. Conversely, if
if a 6 V and if a-1 G V, then h(a) ф 0, for, otherwise,
Л(1) — й(аа-1) = Л(д) А(д-1) — 0,
which implies that h and, therefore, f is trivial. Hence, the
kernel, P, of h is the set of non-units of V, and h decomposes into
the product of the following maps:
viP^h{V)^F,
where к is the canonical map, h'(a + P) = h(a), and i the
injection map. It is now clear that h = <f> on V up to an isomor-
phism. We now extend h to all of К by mapping a into °° if
а ф V. Then h = ф up to an isomorphism, and the proof of the
extension theorem has been completed.
5. Integral Closure
As mentioned before, our main application of the extension
theorem will be to the extension of valuations. However, because
of its importance in number theory, we shall digress in this
section in order to give an application of the concepts introduced
in this chapter and also of the extension theorem to the concept
of integral closure.
Let A be a ring with an identity element and let A be a subring
of the field A. We shall first give the customary definition of the
integral closure of A in K, i.e., we shall give a definition not using
the notion of places, and then shall show the relation of this
concept to that of places.
Definition 5.1. An element <x e К is called integral over A if ex
satisfies an equation of the form
4- • • • 4* an = 0
5. Integral Closure
89
where the ate A. The totality of elements of К which are integral
over A is called the integral closure of A in K.
Clearly any element of A is integral over A. If A = Z and
К = C, an element a which satisfies such an equation is called an
algebraic integer.
We let S denote the set of all places of К which are finite on
A, i.e., 5 is the set of all places of К whose associated valuation
rings contain A.
Theorem 5.1. If a g К is integral over A, and if ф e S, then
ф is finite on a.
Proof. Suppose <£(a) — oo. Then$(l/a) = 0.
However, a satisfies the equation
a" +	+ ••• + йп = 0,
where the a{ e A. Divide by a"; then
1 + a±~ + ••• an— = 0,
and apply the place </> to this. We then have
<£(D = 0,
a contradiction.
The theorem states that any place of К which is finite on A is
finite on any element of К which is integral over A. We shall now
establish the converse of this theorem. Actually, we shall prove
a slightly stronger theorem.
Denote by So the set of those places фе S whose kernel in A is a
maximal ideal of A.
Theorem 5.2. Let <xeK, and suppose Ф °° for any
ф e >S0 . Then a is integral over A.
Proof. We note first of all that once the theorem has been
proved it follows that </>(a)	00 for any ф G S by Theorem 5.1.
We now proceed to the proof.
90 III. Valuation Rings, Places, and Valuations
If a = 0, then a e A and is, therefore, integral over A. Thus, we
may assume that a^O. Consider the ring Аг = Л[1/а]. The
theorem will be proved if it can be shown that 1/a is a unit of
Аг , for this would imply that «g Аг = Л[1/а], which means
that
“ = ao + fli —h " + at ~t >	(5-1)
а	а
where the e A. Multiplying (5.1) by a‘ yields
od+1 — e0«f —	— at — 0,
which implies that a is integral over A.
We shall now assume that I/а is not a unit of At and shall arrive
at a contradiction. If 1 /а is not a unit of Ar, then the principal
ideal (1/а)Лх is not Аг. Thus, there exists (see appendix) a
maximal ideal M of Ал such that
-A! CM.
a
Let us consider the canonical map
Ar AJM.
Since M is maximal, A-JM is a field, which we can inject into its
algebraic closure. In this fashion, we obtain a homomorphic
mapping of the ring Аг into an algebraically closed field. More-
over, the homomorphism is nontrivial since M =£ Ал . By the
extension theorem of the preceding section, we can extend this
homomorphism to a place, ф, of K. Since ф is finite on , it is
finite on A, Therefore, ф e S. Now, (1/а) e M, so <£(l/a) = 0. and,
consequently, </>(«) = °°. This would already yield the desired
contradiction had we used in the hypothesis of the theorem the
set S instead of the set So . Thus, to complete the proof, we must
just show that ф G So .
The kernel of ф in Ar is, of course, M which is a maximal ideal
of Ax . The kernel о£ф in A is A n M — N, and we must just show
that TV is a maximal ideal of A in order to establish that ф G So .
5. Integral Closure
91
We shall show that W is maximal by showing that AjN is a field
(see appendix). Thus, suppose that a 6 A and that а ф N. Then
a e Au and а ф M. Since M is a maximal ideal of Ar and .since
а ф M, a must have an inverse in Аг modulo M, i.e,,
«(*o + *4+- +	= 1 (modM), (5.2)
where the bi e A. However, 1 /а = 0 (mod M), so (5.2) reduces to
abe == 1 (mod M),
i.e., ab0 — 1 e M, but, clearly, ab0 — 1 6 A\ whence, ab0 — 1 eN.
Thus,
ab0 == 1 (mod TV),
where boe A. This shows that A/N is a field and completes the
proof of the theorem.
It follows immediately from Theorems 5.1 and 5.2 that the
integral closure, A, of A in К is a ring, for A is precisely the set
of elements of К which are finite on all places of S.
Next, we show that the word “closure” is justified, i.e., we
wish to show that the integral closure of A in К is precisely A.
Let 5 denote the set of all places of К which are finite on A. If
G S, then, by definition, </> is finite on A, and, therefore, on A, so
</> G 5. Conversely, if G S, then <f> is finite on A since A is the
integral closure of A. Thus, ф s •?, and we have >S == S. To see,
therefore, if an element belongs to the integral closure of A in K,
we must see if it is finite on any ф G S. It follows immediately that
the integral closure of A in К is A.
Finally, we note, in view of our characterization of the integral
closure by means of places of 5 and by the interconnection
between places and valuation rings, that A, the integral closure of
A in K, can be expressed as follows:
А — П V,
where the intersection is taken over all valuation rings of К which
contain A.
CHAPTER IV
Normed Linear Spaces
1.	Basic Properties of Normed Linear Spaces
This chapter will be devoted to certain concepts and results
mainly from analysis which, in conjunction with the results of
the previous chapter, will enable us to tell the complete story on
the extension of valuations. It is possible to proceed in a spirit
much more algebraic and elementary in nature. For example,
instead of introducing Banach algebras and giving an analytic
proof of the Gel’fand theorem, we could instead present the
elementary proof, due to Tornheim [20], that the only
normed field over the real numbers, R, is either R itself or C,
or we could present the more special result of Ostrowski [17].
We choose the present mode of treatment because it affords
us an opportunity to present to the reader a number of
concepts which play an extremely important role in mathematics
today, which happen to have applications to the specific problems
we are concerned with, but have many deep and beautiful
applications far beyond the narrow use we shall make of them.
We first give a rather general definition of a normed linear
space.
Definition 1.1. Let A be a field with a rank one valuation j j.
Let X be a linear (vector) space over k. X is called a normed
linear space over k if there exists a mapping, || ||, called norm,
where
II W-X-^R,
and is such that
(a)	For x e X, || x ||	0	and — 0 if and only if x — 0;
(b)	for x e X and a e A, || ax || == | a 1|| x ||;
(c)	for x, у e X, || x + у || < || x II + ||^ ||.
92
1. Basic Properties of Normed Linear Spaces 93
To avoid special cases in the sequel, we shall always assume
that X Ф {0}. The case X = {0} is always trivial to handle.
Before considering some basic consequences of the definition,
we shall consider a number of examples.
(1)	Take X = k — Q with the usual absolute value onQ as the
valuation and as the norm.
(2)	Take X — Qv, k —Q with the />-adic valuation as the
valuation on Q and its extension to QP as the norm on X.
(3)	Take X = C[a, 6], the set of all real valued continuous
functions on the closed interval [a, 6]. Let k — R, with the usual
absolute value, and define, for x e X, || x || — тах1б[о>|)] ( x(t) |.
(4)	TakeX = C", complex euclidean я-space. Let k = Cwith
the usual absolute value, and define, for x — (eq ,	,..., a.n) e Cn,
|2 + ••• + | «n |2)1/2.
The verification of the conditions (a), (b), (c) of Definition 1.1
in all the examples is quite direct and will be left to the reader.
Let X be a normed linear space. For x, у e X, define
d(x, y) = || x - у ||.
Clearly
(1)	d(x,y) > 0 and equals 0 if and only if x — y;
(2)	d{x,y) = d{y, x),
since || x — j || = || ~(y — x)|| = | — 1 11| j — x|| = ||y - x|);
(3)	d(x, г) d(x,y) 4- d(y, я),
since|| x -z|| =||(x -у) + (у - г) ||	||x — y|| + ||y -г||.
Thus X, d is a metric space, and we have that any normed linear
space is a metric space, and consequently, a topological space. We
can, therefore, introduce the customary topological concepts into
such a space in terms of the metric. For example, the sequence
{xn} of elements of X converges to x G X, denoted by xfl —► x,
if d(x, x„) —* 0, or, in terms of the norm, if || x — xn || —>• 0. Ъпе
can also speak of open and closed sets, compact sets, etc.
94
IV. Normed Linear Spaces
The notion of Cauchy sequences is introduced in the usual
fashion, namely, the sequence {xn} of elements of X is called a
Cauchy sequence if, for every real e > 0, there exists an integer,
N, such that d(xn , xm) < e for all n, m > N.
Finally, we note that one can also consider infinite series of
elements in X.
Definition 1.2. If every Cauchy sequence of the normed linear
space, X, converges to an element of X, then X is called a
Banach space, or a complete space.
Examples (2), (3), and (4) are examples of Banach spaces,
while (1) is not a Banach space.
We now draw some simple consequences from the definition of
a normed linear space.
Theorem 1.1. Let X be a normed linear space. Then
(a) the operation of vector addition in X is continuous;
(i) the norm mapping is continuous.
Proof. Let x, y, x0 , y0 e X. Statement (a) follows immediately
from the inequality
II (* + У) - (*o +b)ll < II* - II + IIJ ~ Уо II.
and statement (b) follows from the easily proved inequality
III *11 - II *0III < II * - *0 Il-
Here, of course, the symbol | | refers to the usual absolute value
function on R.
Let X be a linear space, and suppose that || and || ||2 are two
norms on X, i.e., two mappings of X into R which both satisfy the
conditions of Definition 1.1.
Definition 1.3. The two norms || ||г and || ||2 are equivalent if
there exist two real numbers a, ft both greater than 0 such that
a II * Hi il * Иг P II * Hi
for all x G X.
1. Basic Properties of Normed Linear Spaces 95
It is easy to see that this relation is an equivalence relation in the
set of all norms defined on X. It is also not difficult to see that
equivalent norms give rise to the same collection of open sets in X.
We now establish the following useful theorem.
Theorem 1.2. Let A" be a finite-dimensional linear space over
a field k, where k is complete with respect to a rank one valuation,
| |. Then any two norms on X are equivalent.
Proof. We first define a very special norm on X. Let xt ,
x2 > •••> xn be a basis of X. Then, if x G X, we can write x uniquely
in the form
n
i=l
where the af g k. Define
|| я Ho = max | «f |,
It is clear that || ||0 is a norm on X, and since the concept of equi-
valent norms is an equivalence relation, it suffices, in order to
prove the theorem, to show that any norm, || ||, on X is equivalent
to llllo.
Let x e X, x = Si=1 <Х{Х{. Then
ft
i=l
< X ia< । X(
i=l
iix iio X iix* ii
i-i
= Д1И1о,
where fl — L"=11| x,• || > 0. Thus, to complete the proof, we
must show that there exists a real number a > 0 such that
96
IV. Normed Linear Spaces
a || x ||0 ^ || x ||. We shall establish this by induction on n, the
dimension of X. If n = 1, let xx 6 X, xt 0. Then, for any
x e X, x = a1xt, oq 6 k, and
II x II = II o^Xj || = | oti III Xj || = || X Но II Xj II = a II x ||0,
where a = || Xj ||. Hence, we are done in this case.
We assume now the inequality for all X of dimension я — 1,
and consider X of dimension n. Let xx , хг, ..., x„ be a basis of
X; also, let M be the subspace spanned by x2, x,, ..., xn_!.
The dimension of M is n — 1, so || || and || ||0 are equivalent on
M. Let уг, yt, ... be elements of M and be a Cauchy sequence
with respect to || ||. Then it is a Cauchy sequence with respect to
II Ho • Write
Ji = “1Л + Vs +	li*n-1 >	» = 1» 2,....
where the ajfe k. The sequences {««},; = 1,..., я — 1 are Cauchy
sequences in k since
I ai« — “я I < II У> — Jillo < J = 1. 2,.... я — 1.
Therefore, by the completeness of k,
lim оь, = a, e k for j = 1,..., я — 1.
I-KO
Set
у = ajX! 4-	+ — 4- «„.iX»-! .
Then it is clear that у = lim ytin the norm || ||0 and, consequently,
in the norm || || since they are equivalent on M. Hence, M is
complete. The metric space x„ 4~ M is isometric to M under the
translation mapping: x -* xn 4- x, and is, therefore, also com-
plete. However, any complete subspace of a metric space is closed,
so xn 4- M is closed in X, and its complement is, therefore, open.
Now
0фхп+М,
for, otherwise,
0 = x„ 4- plXl 4- "• 4-	,
1. Basic Properties of Normed Linear Spaces 97
where the 6 k, which would contradict the fact that the xt
form a basis of X. Since Оф xn + M, and since the complement
is open, there exists a real number yn > 0 such that for
x e xn + M
II * II = II * - 01| > y„ ,
i.e.,
II <Х1*1 + - + “я-l *П-1 + *n II > Vn	(1-1)
for arbitrary a, G k. Let x G X, and let
X =	+ OjX2 + -• + Vfl •
We claim that
IIx II ~ II “i*i 4" “2*2 +	4~ ®n*« II + Уп I <*n !•	(1'2)
Equation (1.2) is certainly true if <xn =0; while, if an 0, then
we have from (1.1) that
||-*1 + •” + —*»-l + X„|| >У«,
II	II
which yields (1.2) if we multiply both sides by | an |.
Similarly, we get
II * II = II “1*1 + ««*2 +---Ь “n*n II > Yi I «. I.
i = 1.....n, and where each y, > 0. These other inequalities are
obtained in an analogous fashion as (1.2) by considering the other
(я — l)-dimensional subspaces spanned by я — 1 basis vectors.
Hence,
II * II = II “1*1 + •" + Мп II > a II * Ho .
where a = naify yt > 0, which completes the induction and
establishes the theorem.
In the course of the proof, we established also the following
result.
98	IV. Normed Linear Spaces
Theorem 1.3. If A is a complete field with respect to the
rank one valuation | and if X is a finite-dimensional normed
linear space over k, then X is also complete (i.e., X is a Banach
space).
Although we shall not need it in the sequel, we note the
following consequence of this theorem.
Corollary. If X is a normed linear space over the complete
field A, any finite-dimensional subspace, M, of X is closed.
Proof. The norm, || ||, on X restricted to M makes M a
finite-dimensional normed linear space over the complete field k.
M is, therefore, complete by Theorem 1.3, and, consequently,
closed.
We note that the corollary is certainly not true for infinite-
dimensional subspaces, namely, in Example (3) following
Definition 1.1 take M to be the subspace of C[a, i] consisting of
all polynomials on [а, й]. M is clearly an infinite-dimensional
subspace of C[a, 5], while 1ft, the closure of M, is C[a, A] by the
Weierstrass approximation theorem.
Definition 1.4. Let X be a normed linear space and let {xn} be
a sequence of elements of X. The series xn is called absolutely
convergent if the series II II converges.
For Banach spaces, we have the following result.
Theorem 1.4. If X is a Banach space, then any absolutely
convergent series, xn . is convergent, and
II X*» IH
n=l 11 n=l
Proof. Let xn be absolutely convergent. Then, for any
« > 0, there exists an integer N such that for all я N and any
t 0
II xn+l II + ' + II xn+t II < «•
2. Linear Functionals
99
thus,
II *n+i 4*+ %n+t II < €>
or
II *n+t - II < e,
where sn — xk . Therefore, {sn} is a Cauchy sequence in X,
and lim sn exists since X is complete. Also,
II + — +*„!!< 11*1 II 4— + II »n II,
so
нм <2ii*ji,
n=l
for any n. Hence,
X*n II Xll*n|l.
Л-1	11	n-1
and this completes the proof.
It is readily seen that one can manipulate with absolutely
convergent series in a Banach space just as one does in the case
case of absolutely convergent series of real or complex numbers.
2. Linear Functionals
We shall assume throughout this section that X is a normed
linear space over k, where k = C or R, with the customary
absolute value, i.e., X is a complex or real linear space. Let
f: X —> k be a linear functional, i.e.,/(ox -f- /Sy) = a/(x) + /3/(y)
for all x, уeX and all a, ftek. We shall also consider sub-
sequently linear functionals which are not defined on all of X
but only on some domain Dt С X, where Df is a subspace of X.
We show first of all that if/ is a linear functional defined on X
and if / is continuous at just a single point of X, it automatically
follows that/is continuous on all of X.
100
IV. Normed Linear Spaces
Theorem 2.1	. If the linear functional f is continuous at some
point x of the normed linear space X, then f is continuous
everywhere on X.
Proof. Since f is continuous at x, we have that for every
sequence xn -> x, f(xn) —> f(x). Let у be an arbitrary point of X
and let {yn} be an arbitrary sequence in X such that yn —> y.
Then
f(yn) =/(Уп -y + x+y~x) =f(yn - у + x) + f(y)
However, y„ — у + x —> x, and, by the continuity of f at x, it
follows that
f(y„ ~ У + x) -+f(x).
Thus,
which completes the proof.
We observe that actually all that was needed for the proof was
the fact that f was additive, i.e., f(x + j) = f(x) + f(y) for all
x, у e X. However, we shall not make use of this fact since we
shall be concerned throughout this section with linear functionals.
Definition 2.1. A linear functional f is said to be bounded if
there exists a real number a > 0 such that | f(x) | a || x || for
all x e X.
Similarly one defines a bounded linear functional/ defined on a
subspace Df of X. For a linear functional, the concepts of
boundedness and continuity are equivalent, namely:
Theorem 2.2	. If/is a linear functional defined in X, then/is
bounded if and only if it is continuous.
Proof. Assume first that / is continuous. If / were not
bounded, then, for an arbitrary positive integer n, there exists an
element xn e X such that
l/fc.) I > » II xn ||.
2. Linear Functionals
101
Let yn = *п/я || x„ ||. Clearly, ||jn|| = 1/n, and, therefore,
yn -> 0. But
which contradicts the assumption that / is continuous.
Conversely, suppose that f is bounded and let a be a real
number such that | f{x) | < a || x || for all x e X. Then, if {xn) is
an arbitrary sequence of elements of X such that xn —► 0,
|/(*n) I < a II xn II — 0.
Thus,/is continuous at x = 0, and, by the preceding theorem,/
is continuous everywhere.
We now introduce the notion of the norm of a bounded linear
functional. We observe that since / is bounded the set of all
l/(*) l/ll x II where x #= 0 is a bounded set of real numbers, and,
therefore has a least upper bound.
Definition 2.2. Let/be a bounded linear functional on X, the
norm of/, denoted by ||/||, is defined as follows:
11/11 = sup-1Ж1.
II x II
It follows immediately that ||/|| is precisely the infinum of the
set of all real numbers a which satisfy Definition 2.1. It is also
clear that
|/(*) I <11/1111*11 for all x G X,
and, if e is an arbitrary positive real number, there exist an
x' e X such that
|/(*')l XII/11-^) II*'II-
The norm of/can also be expressed as follows:
II/II = sup I f(x) |.
(2-1)
102
IV. Normed Linear Spaces
This is readily seen to be true, for if || x || < 1, then
l/(*)l <11/1111*11 <11/11.
so sup|W|<11 f(x) | < II/JI- If e > 0, then there exists an x' e X .
such that
!/(*')!> (ll/ll ~ <011 *'II-
Let — x'l\\ x' ||. Then || || = 1, and
l/(*i)1 = iTZHi I Л*') I > liVii (ll/ll - «) II *' II = ll/ll - <;
Ii * II	II л II
therefore,
sup.l/W I >/(*i) I > ll/ll —
||«II<1
Since c is an arbitrary positive real number, we have
sup^u^j |/(x) ( > ll/ll, and this completes the proof of (2.1).
Actually, the proof shows a little more, namely, that
ll/ll = ^l/(*)|.
We note that in the finite-dimensional case, every linear
functional on X is bounded. For let/be a linear functional defined
on X and let хг , x2,..., xB be a basis of X. Then, if x€ X,
x = a,x,, uniquely. Then
/(*) = X “</(*»)>
so
i/(*)i <2>.-|1Ж)|
t-l
< Н*Н0£|/(*.) I
i-1
= «II* llo.
where a — Zt=1 | /(xf) |, and || ||0 is the special norm used in the
proof of Theorem 1.2 of the preceding section. Thus, we have
2. Linear Functionals
103
shown that f is bounded in the norm || ||0, and, since any two
norms on X are equivalent by Theorem 1.2, we have established
the contention.
This is certainly not the case for X infinite dimensional.
For example, consider the subspace M of C [0, 1] consisting of
differentiable functions, and define f:M-+ C, by/(x) = x'(l).
/is certainly linear, but for xn(t) = tn,
/(*n) = <(1) =
from which it follows that / is not bounded.
The next notion we shall introduce is that of a sublinear
functional.
Definition 2.3. Let X be just a linear space over C or R. The
real-valued function, p : X —► R, is called a sublinear functional if
(1) p(x + y) < p(x) + />(y) for all x, у e X\
(2) p(<xx) = ap(x) for all real a > 0 and all x e X.
If, in addition, Xх) 0 for aU хе X, thenp is called a convex
functional.
Our aim now is to establish the important Hahn-Banach
theorem. In order to do this, we shall establish first the following
lemma from which the theorem will follow readily.
Lemma. Let M be a subspace of the real linear space X,
Suppose that p is a sublinear functional defined on N, which is the
subspace spanned by M and some x0 f M', also, assume that/ is a
linear functional defined on M such that
/(x) < p(x) for all x 6 M.
Then/ can be extended to a linear functional £ defined on N such
that
g(x) Xх) f°r all x e N.
104	IV. Normed Linear Spaces
Proof. We first note that any element yeN can be written
uniquely in the form
у = x + ax0
(2.2)
where x e M and a e R, for if
4- axx0 == x9 + ajX,
xx, x2 G M, ax , a2 G R, then
Xj x2 = (a2	<*j) x0 ,
which implies, if aj — ax 0, that xoe M, a contradiction.
Hence, <xx = a2, and then xx = xt, so the representation (2.2)
is unique.
Next, let xx, xs be arbitrary elements of M; then
f(xx) -/(x2) = /(xx - x2) sC f>(*i - *2) = X(*i + *0) - (*2 + *0))
Hence,
-p(-*a - *0) -/(^2) < P(xi + *0)	(2.3)
Since xx and хг are arbitrary elements of M, it follows from (2.3)
that
a = sup (-p(-x - x0) -f(x))
and
b = in£(X* + *0) - f(x))
xeM
are finite. Moreover, a < b. We now choose c to be any real
number such that a < c < b. Then for arbitrary у = x + axoe N,
we define
£(y) = f(x) + ac.
(2-4)
2. Linear Functionals
105
g is clearly a linear functional defined on N, since for
Ji = xi + “ixo > andyt = Xi + in N,
g(yi + Js) = /(*1 + Xt) +	+ Ojj)
=	+ Cal +/(*») +
= g^i) + £(Уз)-
Similarly, g(fiy) = figty) for /? e R. Also, g extends f since, for
x 6 M, a = 0 in (2.4), and g(x) = f(x). Thus, to complete the proof
of the lemma, we must show that g(y) p(y) for all yeN, i.e.,
we must show that
f(x) + ас < p(x + ax0)	(2.5)
for all x e M and all a e R. Equation (2.5) is true for a = 0 by the
hypothesis, so we may assume that а ф 0. Now, since c was
chosen between a and b, we have
-p(-x - xe) -f(x) < c </>(* + *o) ~f(x)	(2.6)
for all xeM. We assume first that a > 0 and replace x by
(1 /a) x on the right-hand side of (2.6). Then

and, since a > 0, we have

or
/(x) 4- ac P(x 4- ox0)
since f is linear and p is sublinear. Thus (2.5) is true for a > 0.
Now suppose that a < 0. We replace x by (1 /a) x on the left-hand
side of (2.6). This yields
c.
106	IV. Normed Linear Spaces
If we multiply by a, we reverse the inequality and get
(-“)/» (- - *o) “c + »/Q-
Again, by the linearity of/ and the sublinearity of p, we have
p(x + ax0) > ac +/(x),
which shows that (2.5) is true also for a < 0 and completes the
proof of the lemma.
Theorem 2.3	. Let p be a sublinear functional defined on the
real linear space X and let M be a subspace of X. Suppose that/
is a linear functional defined on M such that /(x) p(x) for all
x 6 M. Then /can be extended to a linear functional F defined on
all of X such that F(x) p(x) for all x e X.
Proof. We denote by S = {&,} the set of all linear functionals
which extend / and which are such that gj^x} p(x) for all
x e Da , the domain of ga . For ga , gp e S, we define gp > ga if gp is
an extension of g«. This clearly gives a partial ordering in S. If
{£„} is a totally ordered subset of S, then the domains D, of the g„
are totally ordered by set inclusion. We define h on U, Dr by
Л(х) = g,.(x) for x e Dr. h is well defined since the set {£,.} is
totally ordered; also h is clearly linear and belongs to S. Finally,
h > any g, in the totally ordered subset. Thus, S' is inductively
ordered, and, therefore, by Zorn’s lemma S has a maximal
element F. F must be defined on all of X for otherwise, using the
lemma, there exists a proper extension of F, but this would
contradict the maximality of F. Thus, F is defined in all of X, and
since Fe S, F extends /, and F(x) C p(x) for all x e X. This
completes the proof.
We want next to consider the situation for complex linear
spaces. For this purpose, we introduce the notion of a symmetric
convex functional.
2. Linear Functionals
107
Definition 2.4. A convex functional p defined on the real or
complex linear space X is called symmetric if p(ax) — | a, IX*)
for all x e X and for all a belonging to R or C.
Now we can state and prove the analog of Theorem 2.3 for
complex linear spaces.
Theorem 2.4	. Let p be a symmetric convex functional
defined on the complex linear space X and let M be a subspace of
X. Suppose that f is a linear functional defined on M such that
|/(x) I С X*) for all хе M.
Then f can be extended to a linear functional F defined on X such
that
|F(x)|^p(x) for all x 6 X.
Proof. Since X is a complex space, it can also be viewed as a
real space. The linear functional /on M can be decomposed into a
real and imaginary part, namely,
/(*) = X*) + »X*). *eM, (2.7)
where g and h are real-valued functions. Moreover, g and h are
real linear functionals on M, i.e.,
g(ax + fry) = ag(x) + frg(y), and Л(ах + fry) = ah(x) + frh{y)
for all x, у e M and all a, fre R. The proof of this follows readily
from (2.7) and from the fact that a, fre R.
Next, we have, by hypothesis, that
X*) l/(*) I X*) for all хе M.
Thus, the real-valued real linear functional g defined on M, which
can be viewed as a real subspace, satisfies the conditions of
Theorem 2.3. Hence, there exists a real-valued real linear
functional G defined on all of X such that
G(x) С X*) for all x e X.	(2.8)
108	IV. Normed Linear Spaces	।
We define	|
F(x) = G(x) - iG(ix).	(2.9)	[
Observe first that, for x 6 M,
«/(л) = «(M*) + l7l(x)) = -A(*) + «(*)»
but if{x) = f(ix) since/is given as linear on M. Now,
/(»*) = g(tx) + l‘A(IX)-
Therefore,
g(tx) = —h(x), хеМ.	(2.Ю)
Let x e M, then
F(x~) = G(x) — t'G(ix)
= Xх) - #(,x)
= g(x) - »(-A(x))
= g(x) + i’A(x)
= /(*),
where we have made use of (2.7), (2.10), and the fact that G
extends g. Thus, F extends/. F is certainly real-linear since G is,
but it is even complex linear since.
F(ix) — G(ix) — iG(—x)
= G(ix) + iG(x);
while
iF(x) = j(G(x) - tG(ix))
= j’G(x) + G(t'x).
Thus, F is a linear functional on the complex linear space X, and
F extends/. Hence, to complete the proof, we must just show that
F satisfies the desired inequality. This is certainly the case if
F(x) = 0. IfF(x) Ф 0, letF(x) = re«.
F(e-<0x) = e-fflF(x) = e-‘°(r?e) = r.
Hence,
| F(x) | = r — F(e~iOx) = G(e_’flx)
2. Linear Functionals
109
since F(e~iex) is real. But
G(e~i9x) < p(e'**x)
by (2.8). Thus,
|F(x)|O(rt) = M
since p is symmetric. This completes the proof.
Now we continue to consider the situation where X is a
normed linear space over C. Let / be a bounded linear functional
defined on a subspace M. We shall choose a special symmetric
convex functional on X and apply the preceding theorem to this
case; namely, define
/•(*) = H/H II* II for all хе A",
where ||/|| is the norm of / defined on M. Then, by definition of
the norm of /,
|/(*)| < ll/ll II *ll = ?(*) forallxeM;
X*+y) = ll/ll II *+у II < ll/ll (II* II + IIу II)
= />(х) + p(y).
p(ax) = H/Ц || ax|| = | a 11|/|| || * || = | a | p(x).
Thus, p is a symmetric convex functional, and |/(x) | p(x) for
all x 6 M. Therefore, by Theorem 2.4, there exists a linear
functional F defined on X which extends /, and is such that
I*!*) I </(*)= ll/ll II*II for all xeX.
This implies that || F ||	||/||, but it is clear that || F || > ||/||
since F extends/. Thus, we have shown the following:
Theorem 2.5	. If X is a normed complex linear space, and if M
is a subspace of X, then, if/is a bounded linear functional defined
on M, f can be extended to a linear functional defined on all
of X with the same norm as /.
110	IV. Normed Linear Spaces
Of course, the theorem is also true for real normed linear
spaces.
As an immediate consequence of the theorem, we have:
Corollary. If X is a normed linear space over C or R, and if
x e X, and x Ф 0, then there exists a bounded linear functional,
F, defined on X such that || F || — 1, and F(x) = || x ||.
Proof. Let M be the subspace of X spanned by x. Then, for
yeM,y —xx, and we define
/(j) = a || x||.
f is clearly linear on M, and f is bounded with norm 1 since
|/O)| = I « 111*11 = Ill'll
for all у e M. Therefore, by Theorem 2.5, there exists a bounded
linear functional F defined on X which extends f, and such that
||.F|| = ll/ll = L SinceF extends/,
F(x) = f(x) = || x ||,
and the proof has been completed.
We see from the corollary that if f(x) = 0 for every bounded
linear functional / defined on X, we must have x — 0.
3.	Banach Algebras
We shall now give a brief introduction to the notion of Banach
algebras and we shall derive several consequences from the
definition. Our main aim in this section is to prove the important
theorem of Gel’fand on commutative Banach algebras which are
fields. This theorem will be applied in the next chapter to the
extension problem for valuations, but it is also the starting point
of the Gel’fand theory of commutative Banach algebras. We refer
the reader to the Bibliography at the end of the book for further
details.
3. Banach Algebras
111
Definition 3.1. A normed algebra over C is a set X which
satisfies the following conditions:
(a)	X is a normed linear space over C;
(b)	X is a ring;
(c)	«(xy) = (ox) у = x(ay) for all x, у e X, a 6 C;
(d)	II a₽y|| <||x|| II j||.
We shall assume throughout that all our normed algebras are
commutative, i.e., xy — yx for all x,yeX, and we shall also
assume that X contains an identity element e. Hence, ex — xe — x
for all хе X.
Theorem 3.1	. Let X be a normed algebra. The operation of
ring multiplication in X is continuous.
Proof. Let x, y, x0 , y0 e X. Then
II - *oJo II = II (* - *o) O' -Уо) + Уо(х ~ *o) + x<& -Уо) II
II x-x„II lly-y0II +||y0IIIIX-x0H+llx0IIIIJ -y0 II,
and the theorem follows immediately from this inequality.
If the normed algebra X is complete with respect to the norm,
i.e., if X is also a Banach space, then X is called a Banach algebra.
Theorem 3.2	. Let X be a Banach algebra.
(a)	If x e X, and if ]| e — x || < 1, then x is a unit of X.
(b)	If | A | > || x ||, where Хе С, хе X, then (x — Xe) is a unit
of Д'.
Proof. We consider the series
e + (e — x) + (e - х)г + ••• .	(3.1)
Since || (e — x)n || <£ || e — x ||n, and since || e — x || < 1, by
hypothesis, the series (3.1) converges absolutely, and, therefore,
converges by Theorem 1.4 of this chapter. Multiply the series
(3.1) by x = e — (e — x). Then we have
e + (e — x) + (e - x)2 + ••• — (e — x) — (e — x)2 — ••• = e
112
IV. Normed Linear Spaces
Thus, the convergent series (3.1) represents the inverse element
of x, and statement (a) has been established.
To prove (i), we write
(x — Ae) = A Q x — ej.
Since Л Ф 0, we must just show that ((1/A) x — e) is invertible.
However,
Therefore, by part (a), ((1/A) x — e) is invertible, and the proof is
complete.
Let us denote by A the set of all elements of X which are units.
Then e 6 A, and, by part (a) of Theorem 3.2, the neighborhood
of e:
N(e) — {xe X\\\e — x\\ < 1}
is contained in A. Let xe A, then x-1 exists, and xx-1 — e.
By Theorem 3.1, the ring multiplication is continuous; hence,
there exists a neighborhood of x, N{x), such that
N{x) x-1 C N{e).
Thus, for all j 6 N(x), yx~x 6 N(e) C A. Hence, jx-1 is invertible,
i.e., there exists a x e X such that ух~гх = e, but then у is also
invertible. Therefore, N(x) C A, which implies that A is an open
set.
Theorem 3.3	. The map: A —► A given by x —> x-1 (where A
is, as above, the set of all elements of the Banach algebra X which
are units) is continuous.
Proof. Let {xn} be an arbitrary sequence of elements of A
such that xn —► x, where xe A. Then x-1xn-*-e. Now, (x_1xn)-1
can be expressed by the geometric series
(x~1xn)~1 = e + (e — x-!xn) + (e - x-!x„)2 + •••.
3. Banach Algebras
113
Thus
e -	= ~(e - *"4) - («~ *~4)2 —-
and recalling Theorem 1.4 of this chapter, we have
II « - x?* II < УII (« - II
k-1
= X ii *-*(* _ *»)* ii
t-1
< X ii x~' и* ii x - x« и*
к—1
This clearly implies that x~*x —> e, or x”1 —> x-1, which proves the
theorem.
Let x 6 X, and let Л 6 С. A is called a regular point of x if
(x — Xe) is a unit of X. The set, a(x), of all nonregular points of x
is called the spectrum of x. Hence, A 6 a(x) implies that (x — Ae)
is not invertible. If A is a regular point of x, then (x — Ae) e A,
and, since the inverse operation is continuous on the open set A,
we see that there exists a neighborhood of (x — Xe) such that all
elements у in this neighborhood have inverses. However, by the
continuity of the function x — ae as a function of a e C, there
exists a neighborhood of A, which maps into the neighborhood or
x — Ae. This shows that the set of regular points of x is an open
set in C. Therefore, <r(x), the spectrum of x, is a closed set in C.
However, if | A | > l| x ||, then (x — Xe) e A by the second part of
Theorem 3.2. Thus, a(x) is also bounded and is, consequently, a
compact subset of C. We wish to establish that a(x) is also non-
empty. In order to achieve this, we shall introduce one more
concept in Banach algebras. We could actually have defined this
notion earlier for normed linear spaces, but we only need it for the
special situation at hand.
Definition 3.2. Let G be a domain in the complex plane and
let x be a vector-valued function: x : G —► X, where X is a
114	IV. Normed Linear Spaces
Banach algebra, i.e., for A e G, x(A) 6 X. x is defined to be
analytic in G if
«W -
Л — Лд
exists for all e G.
Let f be an arbitrary bounded linear functional on X, and
suppose that x(X) is analytic in G. f(x(XJ) is an ordinary complex-
valued function on G, and it is analytic on G in the usual sense,
for, since/ is bounded,/ is continuous; whence,
Л -- Лф
exists also for all e G.
Now, we shall establish Liouville’s theorem for analytic
vector-valued functions.
Theorem 3.4	. If x(X) is analytic in the entire complex plai
and bounded, then it is a constant.
Proof. Let / be an arbitrary bounded linear functional on X.
Then, as noted above, /(x(A)) is analytic in the entire complex
plane also. Moreover,
|/(*Ю)| <II/IIHA)II,
and, since || x(A) || is bounded, | /(x(A)) | is bounded. Thus, by the
ordinary Liouville theorem, /(x(A)) is a constant. Hence, for
arbitrary complex numbers a, /8,
/(*(«)) =/(х(Д)),
or
/Ma) - x(?)) = 0,
where/ was any bounded linear functional. Thus, by the corollary
of Theorem 2.5 of this chapter, we must have x(a) — x(ff) — 0,
for any a, ft e C. Therefore x(A) is a constant, which is what we
wanted to prove.
3. Banach Algebras	115
Now, let x be a fixed element of X. We consider the vector-
valued function
x(A) = (x - Ae)-1.
Suppose that Ax and As are regular points of x. Then
«(Ai)-1 x(A2) = (x - At«) x(A2)
= [(* - *2«) + (As - AO e] x(A2)
= < + (A, - Аг) х(Аг).
Hence, x(A2) = x(A2) + (A2 — Aj) x(Aj) x(A2), or
*(A,) ~	= (As - AJ x(Ax) x(\).	(3.2)
We have also seen that the regular points of x form an open set
in C, and we can now, with the aid of (3.2), prove the following:
Theorem 3.5	. The vector-valued function x(A) = (x — Ae)-1
is analytic ip the set of all regular points of x.
Proof. For arbitrary distinct А, Ад in the set of regular points
of x, we have, using (3.2), that
Л — Ад
As A —f Ад , x(A) -> х(Ад) by Theorem 3.3 and by the continuity
of x — Ae as a function of A. Thus,
which completes the proof. We can now prove that, for x e X,
a(x) is not empty.
Theorem 3.6	. If X is a Banach algebra and if x e X, then
<r(x), the spectrum of x, is a nonempty set.
Proof. Suppose <r(x) is empty. Then the set of regular points
of x coincides with the entire complex plane, and (x — Ae)-1
116
IV. Normed Linear Spaces
exists for all A g C. This function is, therefore, analytic in the
entire plane by Theorem 3.5. Furthermore, for А ф 0, e - A~2x —e
as | A | -> oo; therefore, by Theorem 3.3,
(e - A-1x)-1 -> e.
Hence,
|| (x - Ae)-i || = | A-i 11| (e - A~»x)-i || -> 0	(3.3)
as | A | —> oo. Thus the function x(A) = (x — Ae)-1 is analytic in
the entire complex plane and is bounded. Using Theorem 3.4, we
get that (x — Ae)-1 is a constant, which must be 0 by (3.3); i.e.,
(x — Аг)-1 = 0, which is absurd since
(x — Ae) (x — Ae)-1 = e.
It is now a simple matter, on the basis of Theorem 3.6, to
establish the Gel’fand theorem.
Theorem 3.7	. If X is a Banach algebra (commutative, as
usual) which is a field, then X is isomorphic to the field of complex
numbers.
Proof. Let x G X, a(x) ,6 <f> by Theorem 3.6. Let A g o(x)
then (x — Ae) is not invertible. However, X is a field so every
nonzero element is a unit. Thus, we must have x — Ae — 0, or
x = Ae. In other words, every element x G X is of the form Xe
for some A 6 C. The mapping, X —»• C, given by Ae —► A now
yields the desired isomorphism.
This completes our short excursion through normed spaces and
Banach algebras. For the most part, we have just proved things
which will be needed for the discussion in the next chapter.
However, we again hope that the trip was worthwhile for the
reader in the sense that he will be motivated to discover more
of the properties and applications of the structures, which we have
considered all too briefly in this chapter. References are, for
example, Naimark [15], Lorch [14], Hille and Phillips [9].
CHAPTER V
Extensions of Valuations
1.	The Extension Problem
In this chapter, we shall be concerned for the most part
with the problem of extending a rank one valuation on a given
field to a finite extension field. We shall see that it is always
possible to do this; however, we shall have to handle the archi-
median and non-archimedian cases separately. If the given field is
complete with respect to the valuation, it will be seen that the
extension is unique, and an explicit formula for it will be obtained.
We shall next consider the number of different extensions which
are possible when the given field is not complete. In the course of
the development, a number of concepts will be introduced and
some results will be obtained which are important beyond just
the extension problem.
We shall begin the discussion with the case of a given field k
with a rank one valuation | | such that k is complete with respect to
| |. К will denote a finite extension field of degree n of k. We shall
show that, if the valuation can be extended to K, the extension is
unique.
Since К is a finite extension field of degree n of k, К can be
viewed as a finite-dimensional vector space of dimension n over k.
Let Xj, хг, ..., xn denote a basis of К over k. We denote the
extension of | | on the complete field k to К again by | |. | |, being
a valuation on K, makes К a normed linear space over k according
to Definition 1.1 of Chapter IV. If x 6 K, we can write
<X = Ot^X^ 4" * ' * 4* an^n	(1*1)
in a unique way where the af 6 k. If we set
|| x ||0 = max | <х( I,
then this is also a norm on K, as we noted in the proof of Theorem
1.2 of the previous chapter. However, by this same theorem,
117
118
V. Extensions of Valuations
|| ||0 and | | must be equivalent. Also, К is complete with respect
to either norm by IV, Theorem 1.3.
Suppose that | x | < 1, then | x |r —► 0, so | xT | -► 0, which
implies, by the equivalency, that || xr ||0 -*• 0. But this means,
writing
XT = <xnxl +	4------F «nXn .
that
lim | aT< | = 0, i = l,2,..., я.
r-wo
(1.2)
Now, the norm of x, NK^(x) = N(x), for x given by (1.1) is a
homogeneous polynomial of degree n in the af with coefficients
in k. Thus, we must have
lim | N(xr) | = 0,
(1.3)
for N(xr) is a homogeneous polynomial of degree n in the ari with
fixed coefficients in k for all r, and then use (1.2). By the multi-
plicative property of the norm and by the multiplicative property
of the valuation, (1.3) can be written as
lim | N(x) Г = 0,
(1.4)
which implies that | N(x) | < 1. We have, therefore, shown that
if | x | < 1, then | N(x) | < 1.
If | x | > 1, then | 1/x | < 1; whence | N(\/x) | < 1, but
N(l/x) = N(x)~r. Thus, j N(x) |-1 < 1, and | N(x) | > 1. Hence,
if | N(x) | = 1, then we must have | x | = 1.
Finally, for xe K, and x Ф 0, let у = N(x)lxn e K. Then
NM _ MM*)) м*г _ t
W N(x") N(x}n ’
which implies, by what we just noted, that | у | = 1, i.e.,
|M*)| = 1
I x” I
1. The Extension Problem	119
or
|N(*)| = Inl-
and
I * । =	(i.5)
for all x Ф 0 in K, but (1.5) is clearly true for x = 0. Thus, (1.5)
holds for all x e К and shows:
Theorem 1.1. If a rank one valuation of the complete field
k has an extension as a rank one valuation to a finite extension
field К of degree n over k, the extension is unique and is given
by (1.5).
In particular, we note that the trivial valuation of a field k
does not have any nontrivial extensions to finite extension fields.
This is certainly not the case for infinite extension fields as we
observed in the example of the rational function fields of I, 2.
Now, we turn attention to the extension problem. Here, as
noted earlier, we shall have to consider separately the archimedian
and non-archimedian cases. We consider first the non-archi-
median case in general, i.e., we let | be an arbitrary non-
archimedian valuation of a field k, not necessarily complete, with
associated place, 95, and associated valuation ring Vk, with units
Uk and non-units Pk . Recalling the discussion in III, 2, we have
| k* |fc ~ k*/Uk , and there is an equivalent valuation with
ordered group k*jUk, such that | aVk | < 1 if and only if
eeP** =P*-{0}.
Next, let К be an arbitrary, not necessarily finite, extension
field of k. We want to show that | |fc can be extended to K. As
noted in III, 4, we may assume
<p : k —F и {<»},
where F is an algebraically closed field. If we consider <p restricted
to Vk , then
Vk-+F,
where F is algebraically closed, and where <p | is nontrivial. We
note here that we may assume throughout that | |t is not trivial
120
V. Extensions of Valuations
since the trivial valuation on k can always be extended to a
trivial valuation on K. Now, by the extension theorem of III,
4, ?> can be extended to a place ф of K. We designate the
associated valuation by | |K, the associated valuation ring by ,
the units and non-units by UK and PK, respectively.
If a 6 Vk, then <p(a) 6 F; whence, ф(а) e F since ф is an extension
of <p\yt- But ф(а)ёР implies that aeVK. Thus VkCVK.
Now, let a G k and а ф Vk, then <p(«) = <», so gs(l/a) = 0;
therefore, ф(1/а) — 0, and ф(а) = «>, i.e., а ф VK. Thus,
Vk = koVK,
(1-6)
and ф\к = <p.
| |K restricted to k gives clearly a valuation on k, and for a G k,
| a |K 1 means that a G k n VK = Vk. Hence, | restricted
to k is a valuation of k equivalent to | |t, the given one. Moreover,
from (1.6) and from III, (1.2), we have
К -Vk = К -ku К -VK,
k—Vk — ko(K — FK),
(Pfc-{0})-‘ = kn(PK-{0})~\
Therefore,
P к — n Pk >
and, consequently,
Uk = k n UK.
We can now embed k*/Uk in K*fUK in an order isomorphic
fashion by mapping:
aUk —► aUK .
The mapping is well defined for, if b ё Uk , then ab(Jk^>- abUK
= aUK. It is clearly a homomorphism, and, if aUK — UK,
then аё UK ok = Uk. Thus, it is isomorphism. Finally, it is
order preserving for aUk < bUk implies that ab-1 ё Pk = PKo k,
1. The Extension Problem
121
so aUK < bUK. Also, | A* |t is order isomorphic to k*[Uk ,
and | К* I* is order isomorphic to K*jUK,
Similarly, the mapping
a + Рц —* a + Pk
is an isomorphism of Ик/Рк into VKIPK.
Thus, the discussion shows that the given valuation | can be
extended to К and that we may view the value group | k* |t as a
subgroup of the extended value group, and the residue class field
Vk/Pk as a subfield of the extended residue class field.
Summarizing just part of this, we have:
Theorem 1.2. If k is a field with an arbitrary non-archimedian
valuation | |, and if К is an arbitrary extension field of K, then | |
can be extended to K.
Now, let us consider a non-archimedian rank one valuation
| | on k, so | k | can be considered as a subgroup of the additive
group of real numbers. Again, let К be an extension field of k.
We want to see if it is possible to extend | | to К so that the
extended valuation is also of rank one. We shall see that this can be
done if A" is a finite extension of k. Before proving this, two
concepts will be introduced.
Let j | be a non-archimedian valuation of k with associated
place <f> and valuation ring Vk , with non-units Pk . Let | | denote
also its extension to К with associated place ф and valuation ring
VK , with non-units PK where К is an arbitrary extension field of
k. Then, as noted in the discussion prior to Theorem 1.2, | k* |,
can be considered as a subgroup of | K* |, and <р(Рк) — Vk]Pk as
a subfield of ф(УК) ~ Vk/Pk •
Definition 1.1. The index, e = [| K* | : | A* |], is called the
ramification index, and the degree,/, of the field ф(УК) ~ VkIPk
over <p(Vk) ~ Vk/Pk is called the residue class degree.
In order to settle the question raised before the definition, we
shall first establish the following inequality between e, f and the
degree n of a finite extension field of k.
122
V. Extensions of Valuations
Theorem 1.3. If К is a finite extension field of degree n of k,
then both e and fare finite, and ef ^n.
Proof. Let Xj, xt,..., xt be elements of VK such that
^(*1), i/r(x2),Ф(х{) are linearly independent over <р(Ил). Also,
let y1, yt,..., yt be elements of K* such that | уг | | k* |,
| y2 | | k* |.| yt | | k* | are distinct cosets of | k* | in | K* If
we can show that the ij elements x,yu , v = 1,i, p = 1,of
К are linearly independent over k, then we shall have that
ij n, which implies that e and f are finite and ef C Now, let
us show that these elements are, indeed, linearly independent
over k.
Let a, 6 k, v = 1,..., i. We claim that
| a-jX! + агхг + ••• + djX,, | = max | a„ |.	(1.7)
This is certainly true if all the a, = 0. If not, suppose
| ar | = max, | a, |, and let	— v = 1,..., i. Then
| b„ j 1, and bj = 1. To establish (1.7) it is clear that we must
prove that
I xi +	+ •" + bixi I = 1-	(1-8)
Clearly, |	4- b%x2 4-	4- Ь(х( |	1 since xx 4- Ьгхг +
4- bjXj e VK. If it were < 1, then
ф(х! 4- 62x2 4- ••• 4- Ь<х() = 0,
which implies that
Ф(х1) + Ф(ьг) Ф(хг) H-----h Ф(ь<) 0(x.) = 0,
or
Ф(х1) + Ф(хг) 4-	4- <p(bi) ф(х() = °>
and this is a contradiction of the assumed linear independence of
^(xj),..., <Д(х,). Therefore, (1.7) has been established.
Finally, suppose that
2w=0>	(1.9)
1. The Extension Problem
123
where the ctfle k. Equation (1.9) can be written as
X Л = 0-
(110)
Now, | j \y„ | = 0 or belongs to | уц | | k* | by (1.7).
Since the | y^ | | A* | are distinct, the nonzero terms of (1-10) have
different valuations. Thus, if some | cr/pc, И Уц | ¥=0, then
| X (X^'*’) yf | = m*x | X c^x' | I I °>
which contradicts (1.10). Hence, all |	| | yfl | =0, i.e., all
|	| = 0. However, again by (1.7),
0 = | X C^X' I = myax I I-
Thus all the cVfl = 0, and the are linearly independent. This
completes the proof.
Now, we can prove:
Theorem 1.4. If | | is a non-archimedian rank one valuation
of k, and if К is a finite extension field of A, then | | can be extended
to a rank one valuation of K.
Proof. By Theorem 1.2, | | on k can be extended to K. We
denote the extended valuation by | | also. We also know that
| k* | C | K* [ and that | k* | C R. Finally, by Theorem 1.3, we
have that [| K* | : | k* |] = e is finite. | |* is clearly an equivalent
valuation of K, and, for а ё К,
I a I' e I A* I c R.
Thus, we have obtained an equivalent valuation with values in R.
Hence, (| a I*)17* (where we extract the eth root in R) is an
equivalent valuation with values in R, and for a e A, it coincides
with| a |. This completes the proof.
There is now just one further matter which we wish to discuss
124
V. Extensions of Valuations
concerning the extension of non-archimedian valuations | | of
rank one on a field k. We have seen, in general, by Theorems 1.1
and 1.2, that, if k is complete with respect to | | and if К is
a finite extension field of k, the extension of | | to К is unique.
Now, suppose however that k is not complete with respect to | |.
In how many ways can we extend | | to К ? We shall first show that
there are only finitely many extensions, and then, in the next
section, we shall determine just how many there are. To show that
there are just finitely many extensions, we shall prove an extended
version of Theorem 1.3 for rank one valuations. Although, the
theorem we shall prove is also true for arbitrary non-archimedian
valuations the proof is much harder, and we refer the reader to
Zariski and Samuel [23] or to Roquette [18] for the details.
Theorem 1.5. Let k be a field with a nontrivial, non-archi-
median rank one valuation, | |, and let A be a finite extension field
of degree n of k. Finally, let | |t, | |, ,... be distinct valuations
of К which extend | [. If q , et,... and ... are the respective
ramification indices and residue class degrees, then ^ekfk n.
Proof. First observe that the valuations | |x, | |2,... are not
equivalent. Let aek be such that | a | Ф 1, and | a |	0. If
If I I? = I b > then, since | |,- and | |, extend | |, we would have
| a = | a |, which implies that a = 1, and | |< = | b , a contra-
diction. Also, in virtue of Theorem 1.4, we may assume that the
| |t- are rank one valuations.
To the given valuation 114, we choose elements xlk, x2k,..., x/ik
and elements ylt , y2k , ..., уе*к as in the proof of Theorem 1.3.
Next, by the approximation theorem of I, 3, we can find for each Л
and each i and j elements , o^j........af*k and elements /3lt,
Pzk . —. Petk such that
< |Л*|*	(LU)
I Pit I. < min | Уп le for all r #= k,
r,t
and
I — xik I* < 1
I xik Is < 1 for all s Ф k.	(L12)
1. The Extension Problem
125
Now,
I Plk I* ~ I У{к ~ (Унс ~ Pik) It = | У^к I*
by (1.11), and, since the yjk were chosen such that the cosets
I У>к It I I are distinct, we have that the | pSk |* (/ — 1.....et)
are in distinct cosets of | fe* | in | K* |. Also, we claim that the
elements ^»(aik), i = 1, ..-,fk are linearly independent over
<p(rt), where фк, of course, denotes the place associated with
| |fc - This follows from (1.12) since | a<t ~ x<t It < 1 implies
that — xik) = 0, so
•At(ait) — l/'t(Xit)'
and the </>*(*«) were chosen to be linearly independent over
<p{Vk).
Vfe shall now show that the elements
«aftt	(1-13)
are linearly independent over k. Once this has been shown, since
(1.13) constitutes ^ekfk elements, it follows that	я, and
the theorem will have been proved. Thus, suppose that
3^ aijkolikP)k = 0,	(1.14)
«J.t
where the aijk g k. We may assume that | eui | = maxM k | aijk |,
and we write (1.14) as follows:
2/ (2/ Ph + 2/ aijkaikfSjk = 0.	(1-15)
i ' i	i,i,k^k
Recalling (1.7) of Theorem 1.3 and its proof, we have
|(2/ Pn I «,n I I Ph Ii
— I «in I I Ph Ii •
126
V. Extensions of Valuations
Therefore, since, as noted, the | j831 are in distinct cosets of | A* |
2 (2 Дл = mfx |(2
I am I I &i Ii •
Next, consider the second sum in (1.15). For k > 2,
and
and
Thus,
I aiik I I alll l>
I «all < 1 [by (1.12)],
1А*11<1Д1111 [by (1.11)].
I aiik I I “tt 11 I Pik 11 <‘l й111 I I Pll 11 •
Therefore,
0 - j 2 aUI&ikftik j — j 2 Л<Л«л) fill
I fllll I I fill 11 •
Hence, | aui | = 0, and it follows that | aijk [ = 0, or aijk = 0
for all t, j, k, which completes the proof.
It follows immediately from the theorem that there can be at
most n extensions of | | on k to K.
As the final consideration of this section, we shall discuss the
problem of extending an archimedian valuation | | on k. Suppose
for the moment that k is complete with respect to | |. Consider the
field A(i) where i is a root of the equation x2 + 1 — 0. If the
valuation can be extended to k(i), we know it is unique by
Theorem 1.1. In fact, for aeA(i), a — a -f- ib, and
I a | = V\ 2V(a) | = Va* + b2.
(1-16)
1, The Extension Problem
127
But it is readily seen that (1.16), indeed, defines a valuation of A(i)
extending the given one, for clearly | a |	0 and = 0 if and only
if a = 0. Also,
10481 = V|N(a^)“| =	= VN& VN(P) = I «I IД I,
and, finally, a straight forward computation shows that

Now, let | | be an archimedian valuation of the field k. Then k
must have characteristic 0 (see I, Exercise 5). Thus, k contains an
isormorphic replica of Q, and | | restricted toQ is equivalent to the
ordinary absolute value. By replacing | | by a suitable power, we
may assume that | | is the customary absolute value on Q. We
designate, as usual, by k, the completion of k. Then we have:
The valuation of £ can be extended uniquely to £(i) as we have
just noted and is a norm on £(»). Also, ^(ij is complete by IV,
Theorem 1.3. Hence, £(j) is a commutative Banach algebra which
is a field. Therefore, £(i) ~ C by IV, Theorem 3.7; whence k is
isomorphic to a subfield of C. Thus, we have shown:
Theorem 1.6. Any field k with an archimedian valuation is
isomorphic to a subfield of the field of complex numbers, and
the valuation of k, viewed as a subfield of C, is a power of the
usual absolute value.
If A is a finite extension field of k, and | | an archimedian
valuation of A, then k, as we have just seen, is isomorphic to a
subfield of C. Since C is algebraically closed, К is also isomorphic
128
V. Extensions of Valuations
to a subfield of C. Mapping К isomorphically into C by an
isomorphism which agrees with the given one on k, gives rise
clearly to an extension of the valuation on k to K. There are
only finitely many such isomorphisms, and, hence, only finitely
many extensions of the given non-archimedian valuation. We shall
go into these matters in more detail in the next section and shall
work out a number of specific examples.
2. The Number of Extensions of a Valuation
Let k be a field with a nontrivial rank one valuation, | |. Let k
denote the completion of k with respect to | |. We first establish
the following preliminary result.
Theorem 2.1	. Let К be a finite extension field of k, and let | |
be a rank one valuation on k. Denote an extension of | | to К again
by | |. Then JC = Kk, where Kk is the composite field formed in
Proof. Clearly Kk is a finite extension field of Я, and, since k
is complete, Kk is also complete by IV, Theorem 1.3. Now, since
К C Kk and since К is dense in /t, we have /t C Kk\ whence
Kk — which completes the proof.
Let K± and K2 be two finite extensions of k, and let a : Kr —► K2
be an isomorphism of Kt onto K2 which leaves k fixed. The
valuation | | in k can be extended uniquely to | |j on and
uniquely to | |2 on K2. We claim, that under these conditions,
that, for any a G Kx, j a |x = | <r(a) |2 , i.e., a takes the valuation of
into the valuation of K2. This fact is clear since, if n is the
degree of these fields over k,
I “ li =	= <1 Ж“)) I = I <<*) Is •	(2.1)
because a is an isomorphism leaving k fixed.
Again, let A be a field with a rank one valuation, | |, and let
К be a finite extension field of k. If A is the completion of k, we
designate by the algebraic closure of A. Since k is complete, | |
2. The Number of Extensions of a Valuation 129
can be extended to k uniquely, for, if a e if, then k(a) is finite of
degree я, say, over k, and we define
I « I =	(2.2)
where the N(oi) = Nfa^.Joc). It is easy to see that (2.2) defines a
valuation on it, which extends | | on k, and is unique. The
uniqueness is clear, while clearly | a | > 0 and = 0 if and only if
a = 0. Moreover, it does not matter what finite extension field
containing £(a) we compute | a | in, namely, if & Э F D H(a),
where F is of degree m over £(a), then computing | a | in F gives
nm,---------
I«I = V| ^|£(«) I,
but NF^(a) =	and the result follows. Now, if а, Де if,
we can compute | аД | and | a. + Д | in £(a, Д) to check the
remaining axioms, and, since | | is a valuation of Ifla, fl), this
completes the proof.
We can extend | | to К if we isomorphically embed К in if by an
isomorphism leaving k fixed, namely, if
Л : К — Кг C I,
where A on k is the identity, then let | |t be the restriction of | |
given by (2.2) to , and, for fleK, define
IД к' = I 4Д) к •	(2.3)
Clearly | |/ is a valuation of К which extends | | on k and is
induced by the embedding.
If
At: A'->X1Cif,
and
A,: K-*-KtC^
are two embeddings leaving k fixed, and if a is an automorphism
of К which leaves ft fixed and is such that aA1 = A2, then, by
(2.1) and (2.3), for Де К
I Д к' = I АГ(Д) к = I <^(Д) к = I 4(Д) к = IД к'.
130
V. Extensions of Valuations
provided that Кг and Кг are extension fields of X; otherwise, we
consider Kji = and K2k = £3. Two embeddings related
as above are called conjugate over ft, and we have shown that
conjugate embeddings induce the same valuations on K.
The converse of this statement is also true, for, suppose
| |/ = | |g' where | |/ is induced by At : К —► Kx and | |3' is
induced by A3 : К —► K2 . Then Aj = AgAj-1 : Кг —* K2 , and Aj
is identity on k since both At and Ag are. Now, we show that Ag
can be extended to an isomorphism of Kxk, the completion of
Кг, onto K2k, the completion of K2, leaving ft fixed. Let
a. G Kyk; then
a = lim otn ,	(2-4)
I
where the an 6 Кг , and where the limit exists with respect to the
valuation | |j. However, since | |/ = | |2', we have
I *з(“п) - Is = I Аз(“п - «m) Is
= I (an — am) Is
I A1 (an ““ ®m) Is
= I V(“n — “m) 11'
” I 11 •
Thus, {A3(an)} is a Cauchy sequence of K2H, and, therefore,
converges to an element of K2f, which we denote by X3(a). If
we also have a = lim , where the /?„ 6 Kt, and where the
limit again exists in the | |3 valuation, then, as above,
I	^s(an) Is ~ I Pn an 11 >
which shows that X3(a) is independent of the sequence in (2.4).
Hence,
A3;
is well defined. It is an isomorphism, for if a. = lim otn , and
P — lim pn , then ap = lim a„/?„ , and
I Ш) - V«A) Is - I Ш) - Л.К) Is,
2. The Number of Extensions of a Valuation 131
but A3(ocn) -> Ag(a), and X3(ftn) -► Aa(/3). Hence, Ag(a/3) = As(a)Aa(/3).
Similarly A3 preserves sums. If A3(a) = 0, then
I «П |1 = I Лз(“п) Is -* 0,
and а = 0. Also, if /3 is any element of K^, then /3 = lim fln
where the finG K2. Let Aa 1Q8n) = a„ . Then
I «n — “m 11 = I Лз(“п) — Is
= I Pn Pm Is >
so {a„} is a Cauchy sequence in Kxk. Hence, a„ —► a e Ktk, and
*a(“n)->^8(“)> but
I ^з(“п)	^(a) |2 = | P-n A3(a) |2 ,
so j8 = A3(a). Thus A, is onto.
Next, we note that A3 is identity on к since Ag is identity on k.
Finally, Ag can be extended to an automorphism of к.
Therefore, we have shown that two embeddings induce the
same valuation of К if and only if they are conjugate over к.
Next, we show that every extension of [ | to К can be obtained
by embedding К in к. Thus let | | be a valuation of К extending
the given one on k. Then, by Theorem 2.1 of this section we have:
Л
k
Since R. is finite over к, R. can be extended to an algebraically
closed field which is isomorphic to к by an isomorphism leaving к
fixed. This isomorphism takes into a subfield of к and takes,
according to the discussion preceding (2.1), the valuation of £
into that of , or, in other words, the valuation of and,
therefore, of К is induced by that of С к.
We summarize all this in the following theorem.
Theorem 2.2	. Let k be a field with a rank one valuation, | |,
and let К be a finite extension field of k. If к is the completion of k,
132
V. Extensions of Valuations
then the set of distinct extensions of | | to К is in one-one cor-
respondence with those embeddings of К in H, the algebraic
closure of ft, which are not conjugate over ft.
We consider now a special case, namely, let К be a finite
separable extension of k. Then К — k(<x). If/(x) is the irreducible
polynomial satisfied by a over k, then over k
f(x} = p1(x)pi(x) -pr(x),
where the p^x) e £[x] and are irreducible. Also, the pt{x) are
distinct since a is separable. If A is an embedding of k(a) in K,
then we have:
where A(a) = a. Since
0 = /(a) = Pj(a) p2(a) •• • pr(a),
0 =/(«') = A(“')Ps(«')
Hence, some />,(« ) = 0, and the embedding determines uniquely,
by the separability, a polynomial рДх).
Conversely, if we choose a polynomial p^x) and a root a' and
form £(ot') within Л, then, since /(a') = 0, A(a') and A(a) are
isomorphic over k. a, therefore, determines an embedding of
k(a) into ^(а') C £ If we take a different root ft' of p,(x), then we get
f(p') which is isomorphic to &(<x) over f since a and /3' are roots
of the same irreducible polynomial over k. These would then
determine conjugate embeddings of k{a) in and hence, as we
know, the same valuation of Л(а).
We also see that
« = 2».-.	(2.5)
i-1
where n = degree of /(x) = degree of К over k — the global
2. The Number of Extensions of a Valuation 133
degree, and я< = degree ofpt{x) = degree ^(a) over R [where a
is a root of />Дх)] = the local degree.
Thus, we have shown:
Theorem 2.3	. If К = k(a) is a finite separable extension of k,
and if | | is a rank one valuation of k, then there are as many
distinct extensions of | | to К as there are irreducible factors of
/(x) over R, where/(x) is the irreducible polynomial satisfied by a
over k, and where ft is the completion of k. Moreover, the global
degree of the extension is equal to the sum of the local degrees.
If К is a finite inseparable extension, then we could decompose
К into a separable extension, K,, and a purely inseparable
extension of K, obtained by adjoining pth roots to K,, where/» is
the characteristic of k. Since K, is a finite separable extension of k,
K, = k(a), and we can apply Theorem 2.3 to K,. Then we note
that | | on K, is uniquely extendable to K, for, if у 6 С К
where is of degree p over Kt, then y” = p e Ks. If | у | is
defined, then | у |” = | p |, and | у | = | p |1/p. Continuing in this
fashion, we obtain uniqueness.
We return again to the situation where К is a finite separable
extension of k, and where | | is a rank one valuation of k. We
shall look at our previous results from a slightly different point of
view. Let E be the least separable normal extension ofk containing
K. Denote by G, the Galois group of E over k, and by U, the
subgroup of G which leaves К fixed. If 1| is extended to K, then we
know by the preceding discussion that there is an embedding Ax
of К into R. leaves k fixed. Let
К Кг C R
Also, we know that | | is of the form | |/, and, for a 6 K,
| a |/ = | Л/a) |t,
where | |i is the unique valuation of determined by | | on R.
But since Aj is an isomorphism of К which is the identity on A,
At is given by an automorphism e G, i.e., |x = At. Thus
I « Ii' = I »'i(a) Ii.
134
V. Extensions of Valuations
where e G. Also, it is clear that any v e G leads to an induced
valuation of E and, hence, of K, for v : E —► E can be extended to
v : Ek —► Ek, and Ek can subsequently be mapped by an isomor-
phism over k into k, so v gives rise to an embedding in k and to
a valuation on E induced by v.
Now, let
К K2 C k
be another embedding, then, as above,
I « Is' = I •'s(a) Is >
where v3 e G. We also saw that, if | |/ = | |2', there exists an
isomorphism
X3: Kj-^Ktk
such that A3 is identity on k, and	= v2 on K. A3 can be extended
to an automorphism
a: Ek-^Ek
over k, i.e., aeS, the Galois group of Ek over k, which by the
theorem on natural irrationalities is isomorphic to the subgroup
of G which has E n k as its fixed field. Now, a has on the same
effect as X3 since a extends X3 , so
	ovj = v2 on K,
or	v^1<r~1v2 =1 on K.
Therefore,	l»l S/g g U,
or,	v2 G av1 U,
and
v2 e Svi U.
2. The Number of Extensions of a Valuation 135
Conversely, if va = avyi, where a e S and ц e U, then, for a G K,
I « Is' = I ”«(“).|s ~ I ^iK01) la
= I OT'i(“) la
= |vi(«) 11 [by (2.1)]
= I “ It'-
We have, therefore, established the following result:
Theorem 2.4	. Let К be a finite separable extension of k, and
let | | be a rank one valuation of k. If E is the least separable normal
extension of k containing К with Galois group G, and if U is the
subgroup of G which fixes K, decompose G into distinct double
cosets: G = U<_i SvtU, where 5 is the Galois group of E/г,over k.
Then there are as many extensions of | | to К as there are distinct
double cosets, and all of these extensions are of the form:
I »'<(«) |«> i = 1, •••> r, where the | |( are the unique extensions of
| | in k to the embeddings of К in £ associated with the v{.
We shall now apply this theorem to obtain a result, particularly
useful in number theory. First, write G as a distinct union of
double cosets: G = U, SvtU. Since
SOSa ViUv^,
we can decompose S into disjoint cosets
S = U ^(S^A?1) .	(2.6)
j
Hence,
SvJJv^1 = U ,
i
or
SViU = и <wtu.
J
We claim that this last union is disjoint, for suppose that
OijViU = oikv{U (j ф k).
136
V. Extensions of Valuations
Then
a7kauvie ’«Чл17 = viU>
so
«ъЧл = w
where p e U. Thus,
= ^r1,
and, therefore,
a^eSn^,
which is a contradiction. Hence, we can write
G = U auvi U (disjoint).	(2.7)
The group of v<(K), where we use the same notation as in the
theorem, is v{Uvf\ while the group of v^K) к is S н vtUvf\Thus
the trace of ^(a), where at 6 K, from v/(jK) £ to £ is given, using
(2.6), by
and the trace of a from К to k, using (2.7), is given by
(“) = S’«*<(«)•
I* ti
Therefore, we have
SK^«) — 2/ ^4<лг)»| _4(a))-
In other words, the global trace is equal to the sum of the local
traces, and, similarly,
or the global norm is the produce of the local norms, where all of
these results have been established under the assumptions of
3. Valuations of Algebraic Number Fields
137
Theorem 2.4. One frequently abbreviates these formulas by
simply writing
and
AGq («) = П^(, («)•
Ik	t Ik
These results hold under somewhat broader conditions than
those of the theorem, but we shall not go into such matters here.
3. Valuations of Algebraic Number Fields—Examples
If К is an algebraic number field, i.e., К = where в
satisfies the irreducible polynomial/(x) overQ, then we can apply
the results of the last section, in particular Theorems 2.2 and 2.3,
along with the results of Chapter I. We know, disregarding
equivalent valuations and the trivial one, that the only valuations
of Q are the ordinary absolute value and the p-adic valuations.
To see how these extend to K, we apply the preceding results.
For example, if | | is the ordinary absolute value, then Q = R,
and/(x) decomposes, in general, over R into linear and quadratic
factors:
/(x) == (x - oq) ••• (x - ar)p1(x) ••• pt(x).
The extensions of | | to К are now obtained by considering all the
non-conjugate embeddings of К in Q = R = C determined by
roots of these polynomials. These various embeddings induce all
possible valuations on К extending | | on Q, and we see that there
are r + t of them.
In the case of a p-adic valuation | jp on Q, the same sort of
observations hold only now Q = Qp, and we must investigate
how /(x) factors over Qp . Each irreducible factor determines an
embedding of К in Qp and induces a valuation on К extending | |p.
We shall now consider some specific examples to see how these
steps are carried out, and to see how the extended valuation is
determined for various elements.
138
V. Extensions of Valuations
Example 1.	Take К = Q(6), where в satisfies x2 + 1 = 0; and
let | | be the usual absolute value on Q. Over jR, x2 + 1 is
irreducible. Hence there is just one extension, | |', of | | to K,
obtained by embedding Q(ff) in Q(i) С C and taking the induced
valuation. Thus,
| a + Ьв I' = | a + Ы | = Va2 + ft®.
Example 2.	Let К — Q(&), where 0 satisfies the irreducible
equation x3 = 3, and let | | be the usual absolute value on Q.
Over R, x3 — 3 factors into a linear factor (x — tyi) and a
quadratic factor p(x) with roots -^Зе and -(/Зе2 in C, where e is a
primitive cube root of unity. There are, therefore, two extensions
| I/ and | |2' of | | to К obtained by embedding Q(6) in 0(^3) С C
and in Q(^3e) С C.
For example,
| 1 - в |/ = | 1 - № | = № - 1.
|l-0|2'-=|l-e<3|.
Example 3.	Let К = Q{0), where в satisfies the irreducible
equation/(x) = x3 — x — 1, and consider the 17-adic valuation,
| |17, on Q. Let ац = 5; then /(aj = 119 = 7 X 17, and
f	Thus, !/(«!) |17 < 1, and Ii? = !• Therefore,
by II, Theorem 4.1, the sequence
«1> > -
of II, (4.1) converges to a root /3 of /(x) in Q17 . The second term
otg, is given by
2 ~ /'(«,)
= 5-H?
74
=	= -63 (mod 17®P),
3. Valuations of Algebraic Number Fields 139
where ? is the valuation ring of | |17 on Q17 . We could clearly
compute Д to any desired degree of accuracy. Now, overQ17 ,
x3 — x — 1 = (x — Д) Xх),
where/>(x) is a quadratic polynomial, which we claim is irreducible
over Q17. In order to show this, we must just show that p(x)
has no root modulo 17, for if, say, p(a) = 0, where aG017,
let a = a (mod 17?), where aeZ. Such an a exists for, if
X°0 — 0, then a3 — a — 1 =0, and clearly «e f. Then
0=p(a) =/>(«) (mod 17?),
and, since />(«) is an integer,
p(a) = 0 (mod 17).
Now, since 0 = 5 (mod 17?),
p(x) = x2 4- 5x + 7 (mod 17?),	(3.1)
and we must show that x2 + 5x + 7 has no root modulo 17. But
x2 4- 5x + 7 = x2 — 12x + 7 (mod 17)
= (x — 6)2 + 5 (mod 17)
= J2 + 5,
where у = x — 6. However, the congruence
j2 = -5 (mod 17)
has no solution since, by the quadratic reciprocity law,
£) = » - (n) - (t) - © - ->
Thus, there are just two extensions of | |17 to К obtained by
embedding
Q(0) ЭД8) C Q17,
and
Q(0) ~ Q(y) C Q17,
140
V. Extensions of Valuations
where у is a root ofp(x) in 017 . Denote the induced valuations by
| |/ and | |j', respectively. Let us compute some specific values
for these two extensions. Consider the element 3 - 28 + 66» eQ(8).
| 3 - 20 + 60»)/ = | 3 - 20 + 60» |„.
But
3 - 20 + 60» s 3 - 10 + 150 = 143 (mod 17?),
and since 17 -f 143, we have
| 3 - 20 + 60» |x' = 1.
If in this computation 17 had divided the number, we would have
to go to the second approximation to 0 and possibly further. For
example, let us compute
|7 + 20 |i'= 1 7 + 20 |x7
7 + 20=17 (mod 17?),
but0 s== —63 (mod 17»?). Thus
7+ 20 =-119 (mod 17»?),
and, since 17» -Г 119, we have
I 7 + 20 I/ = | 17 |„.
We consider next the valuation | |2', and compute 11 + 0 + 0» |2'.
We denote again by | |17 the unique extension of | |17 to Q17(y).
Then
I 1 + 0 + 0»|,' = | 1 +y +y» |„.
From (3.1), we have
у» = -5y _ 7 (mod
Thus,
1 + у + У» = — 4y — 6 (mod 17?),
and | 4y + 6 I,, = | 2y + 3 |I7 .
3. Valuations of Algebraic Number Fields 141
Now, the norm of 2y + 3 is, denoting the other root of p(x)
in 0i7 by y'.
N(2y + 3) = (2y + 3)(2y' + 3)
= 9 + 6(y + y') + 4yy’
s 9 - 30 + 28 = 7 (mod 17?).
Therefore,
I2y + 3|17 = VjTj7= 1;
hence, also
11 + e + v i; = i.
If we had obtained a number divisible by 17, then, as in the
preceding computation, we would have to go to higher approxi-
mations.
Example 4. Let К = Q(0), where 0 satisfies the irreducible
equation f(x) = x® — 2x + 2, and consider | |a on Q, For
eq = 3, we get |/(ax) |гз < l^and ]/'(«!) |a = 1. Similarly for
cij = 9 and otj = 11. Hence, we get three distinct roots a, j8, у in
Qa such that
<x = 3	(23?)
/S = 9	(23?)
у =11	(23?).
Thus, there are three extensions of | |a to K.
Example 5. We now consider К = Q{6), where в satisfies the
same irreducible equation /(x) as in Example 4, namely,
/(x) = x® — 2x + 2, but we consider | |2 on Q instead of | |a as
in Example 4. If we take = 0, then/(ax) = 2, and |/(ax) |8 < 1.
However, /'(ax) = —2, and |/'(<xx) |2 < 1, while dx = so
I <ZX 12 > 1. Therefore, neither version of Newton’s theorem in
Chapter 11 can be applied. If we take ax = 1, then |/(aj) |® = 1,
and, again, we can not apply Newton’s method to see if /(x)
factors over . Instead we argue as follows:
0® - 20 + 2 = 0;
142
V. Extensions of Valuations
hence, if | | denotes an extension of j |2 to K, we see, first of all
from this equation, that | 6 | < 1. Secondly, we see that there
must be more than one dominant term, for we know, in general,
in the case of non-archimedian valuations that if | a | < |/? |,
then | oc H- Д | == | j8 |. Since | 0 | < 1, | 20 | < | 2 |, and we
must have
I 0 I4 s = | 2 | = | 2 |2,
or
I 0 I - I 2 I*/*.
This implies that the ramification index for this extension is at
least 3. However, the degree of Q(0) over Q is 3. Thus, we get
that there is just one extension of | |2 to 0(0) with ramification
index 3 and residue class degree 1.
We leave the examples now, but we note that the computations
performed here are particularly important in number theory. One
is concerned there, among other things, with the factorization of
primes of Z in the ring of integers of algebraic extension fields, and
this is intimately related to the extension of a given p-adic
valuation. We also observe that the two cases considered in
Examples 4 and 5 behaved quite differently. This can be traced to
a consideration of the discriminant of the equation. In general,
for the cubic equation x3 + px -|- q, the discriminant
D = —4p3 — 27q2, which in our case is —4 X 19, so 2 | D,
while 23 -Г D. These considerations are related to the ramifi-
cation problem for primes in number theory, but we cannot go
into these matters here.
4. Discrete Valuations
We shall briefly discuss in this section some properties of
discrete valuations. Many of the properties will be just extensions
of results already obtained for the p-adic valuations, which are
examples of discrete valuations. We assume throughout that k is a
field with a non-archimedian valuation | |.
4. Discrete Valuations
143
Deflnition 4.1. The valuation | | is called discrete if its value
group | A* | is an infinite cyclic group.
If | | is a discrete valuation of k, then | k* | is an infinite cyclic
group and is, therefore, order isomorphic to Z. Hence, there
exists л ire k such that | ir | <1, and | ir | generates | k* |. It is
clear that | ir | is the maximum of all | a |, a e k, such that | a | < 1.
Given any a e k*, a yt 0, there exists an integer, as in the case
of the p-adic valuations, called the ordinal of a and denoted by
ord a such that
| a | = | ir |ori»
Thus, a = 7rorde€, where | e | = 1, so e is a unit.
For a = 0, we take ord a = °°. Then
| a	|	=	1	о	ord	a	=	0,
| a	|	<	1	о	ord	a	>	0,
| a	|	>	1	о	ord	a	<	0.
We also have that
ord ab = ord a + ord b,
and
ord (a + b) > min (ord a, ord b).
If V is the associated valuation ring with P, the maximal ideal
of non-units, it is clear that
P = {a | ord a > 1} = irV.
Aho, ir is a prime element of the ring V, for, if ir = «jOg where
аг, a2 6 V, then, since | ir | < 1, either | 04 | < 1, or 10^ | < 1.
If, say, I otj, I < 1, then a.1 e P — irV, and ir | , but cq | ir;
whence, it is a prime.
Now, let К be a finite extension field of k, and let | | be extended
to A. We shall denote the extension also by | |, and we claim that
this extended valuation in К is also discrete, namely:
Theorem 4.1	. If A is a finite extension field of k, and if Ц is a
discrete valuation of k, then any extension of | | to К is also
discrete.
144
V. Extensions of Valuations
Proof. We know by Theorem 1.3 of this chapter that if n
is the degree of the extension, ef я where e is the ramification
index and f, the residue class degree of the extension. Then
| K* |‘ C | A* |.
Thus, we have a mapping
given by | a | —> | a |* where a g K*. The mapping is an isomor-
phism into | k* |, for it is readily seen to be a homomorphism,
while, if | a |e = 1, then it follows immediately since | k* | is
ordered that | a | = 1. Thus, the ordered group | K* | is
isomorphic to a subgroup of the ordered infinite cyclic group
| k* |, and, therefore, must be infinite cyclic also. This completes
the proof.
Again, we assume that if is a finite extension of k and that | |
is a discrete valuation of k which has been extended to K. We
denote the extension also by | |, and we let e be the ramification
index of the extension. Finally, let | n | be the generator of | k* |
with | тг | < 1, and | П | the generator of | K* | with | П | < 1.
Since | tt | < 1
тг = e77",
where я is a positive integer and where e is a unit of K. However,
| K* | = [| П |], the cyclic group generated by | П |, and
| A* | = [| тг |]> ^e cyclic group generated by | тг |. Now,
|л*1 = [Ы] = [1т«о
[| К* | : | ^* |] = я,
but [| K* | : | k* |] = e. Therefore, e — n, and
тг = e/7‘.	(4.1)
Now, if aek, then, of course, a also belongs to K, and we can
compute its ordinal with respect to тг, denoted by ordt a, or its
ordinal with respect to П, denoted by ordxa. Equation (4.1)
shows that
ordjf a = e ordfc a.
4. Discrete Valuations
145
Next, we simply note that the extension of II, Theorem 2.1 is
valid for discrete valuations; namely, if k is a complete field with
respect to the discrete valuation | |, then every aek can be written
in the form
OQ_
« = X “^>
n
where n = ord a, and where the e V, the associated valuation
ring. The proof is essentially the same as that in Qv and need
not be repeated.
We now wish to show that Theorem 1.3 of this chapter can be
improved if k is a complete field with respect to a discrete
valuation.
Theorem 4.2	. Let К be a finite extension field of degree n of k,
where k is complete with respect to a discrete valuation, j |,
then ef = n.
Proof. We adhere to the same notation used prior to the
statement of the theorem. Let , x2, ..., xf be elements of the
valuation ring VK associated with | | on К such that ф(х1),
Ф(хг),..., </<(»/) is a basis of к) over <p(VK), where ф is the
associated place of | | on K, and <p and are, respectively, the
associated place and valuation ring of | | on k. We wish to show
that every element of К is a linear combination of the elements
xjli, i = 1....f, j — 0, 1, ..., e — 1 with coefficients from k.
Since the cosets
| ** |, | П | | k* |.| П'11|**|
are distinct, we know, by the proof of Theorem 1.3 of this
chapter, that the elements xf!* are linearly independent over k.
Hence, if we can prove that every element of К is a linear сопь
bination of these elements with coefficients from k, then they will
form a basis of К over k, and, consequently, ef = n. Actually, we
shall prove more; namely, we shall show that every element of
146
V. Extensions of Valuations
V? is a linear combination of the xjl’ with coefficients from
Ffc, i.e.,
VK = J V*JV.	(4.2)
i.i
For suppose that (4.2) has been established. Then let aeK.
Select an a 6 k* such that | aa. |	1. Such an a always exists, for
example, some sufficiently higher power of ir will work. Then
i.i
which implies that
ae^kxJV.
ij
Thus, we must just prove (4.2).
Consider 1тг(П* |, where i = 0,1,2,..., and j = 0, 1,..., e — 1.
By (4.1) we have
| rflV | = | П™ |,
but clearly ei + j ranges over all non-negative integers, so
| тг‘П> | yields all values of j | on Vx. Now let a G VK, a 0.
Suppose that
| a |	| rr‘IV |
for some i = 0, 1, 2, ... and some j = 0, 1,..., e — 1. Set
0 = ot/irTT’. Then | /3 |	1, so 0(0) 6 0( VK). Therefore,
0(0) = 0(ai) 0(*i) + 0(аг) 0(*г) + •" + 0(az) 0(*/),
where the a, g Vk . Hence
0(0) = 0(аЛ +	+ - + atxt),
or
0(0 - («i^i H------h a/Xf}) = 0.
But this means that
10 — (аЛ + — + afxf) | < 1,
4. Discrete Valuations
147
or, recalling the definition of Д, that
I ® — («1*1 + " + afxf) "‘л* I < I ^П1 |.
Let
a —	+ ••• + afxf) ir*!!1 = y.
We have, therefore, shown that, if | a | ^ | ir*!!1 |, there exist
elements ax,..., af in Vk and an element у e VK such that
a. = (ap^ + ••• + afxt) ir*!!1 + y,
with | у | < | ir*!!1 |.
Now let a be an arbitrary element of VK, so | a |	1. By the
preceding result, we can write
“ = aooixi 4~ аоогхг + ‘ + «оо/*/ + ai >
where the Vk and where | oq | < 1, so | at | < | 77].
Similarly, considering ax, we have
“i = («011*1 +------h «oir*/) П + “a .
where ] oig | < | П |, so | og |	| Пг |. Similarly,
®4 ~ («021*1 4“ ’' ’ 4“ «02/*/) 772 4" ^3 >
where | a3 |	| П3 |. Continuing, we get down to
a«—1 “ («08—1,1*1 “h + «Oe—1./*/) 77е 1 “Ь	>
where | ae | <2 | Пе | = | ir |. Hence,
a« = («101*1 4" ’'" 4" «10/*/) 7r 4- a8+l ,
where | aefl | < | irl! |, so
“«+1 ~ («111*1 4-	4- «Ц/) rr77 + ae+2 >	’
where | ae+s |	| л-П2 and so on.
Performing successive substitutions, we see that a can be
148
V. Extensions of Valuations
written as a certain finite sum plus a small remainder term,
namely,
« = X + •" + anfxr) + у,
i.i
where the sum is finite and where | у | is arbitrarily small. Thus,
“ = X	+ ' +e«/xr)ni <4-3)
i.f
where i = 0, 1, 2, ..., and j = 0, 1, e — 1. We can write
(4.3) in the form
« = X ((X X1 + (X ai^) *2 + "•+ (X а«/7Г‘) Xf)n1’
j=0 " i	t	' i
the E, a^ir* converge since the aijk e Vk and | a^ir* | —► 0. In
fact, E$ а(1ктг* e Vk , since, if sn denotes the «th partial sum,
| sn | <2 1; whence, | E, аик1г{ | — lim | sn | <2 1, and this shows
that (4.2) is true and completes the proof of the theorem.
We saw in II, Theorem 1.4 that if | | is a non-archimedian rank
one valuation on k, then | k | = | k |, where k, as usual, denotes
the completion of k. We note that the same is true for the residue
class fields, namely, the residue class field of k is the same as that
of k. For, if <p is the associated place, and if a G k, | a | Cl, then
since k is dense in ft, there exists an a G k such that | a. — a | < 1,
or a = a ft, where | /8 | < 1. Hence 99(a) = 99(a), which
establishes the contention.
Next, we observe that Theorem 4.2 is not true, in general, if
I I is not a discrete valuation even though the field k may be
complete. This may be seen as follows: We start off with the
field Q and the 2-adic valuation | |2 and form the completion, Q2 .
Now, form Q2(\/2). The valuation can be uniquely extended to
Q2(V2) and is given by
I«I = viWh
for cxgQ2(V2). Also, Q2(\/2) is complete since it is finite over
4. Discrete Valuations
149
Q2, and, by Theorem 4.1, the valuation on Q2(y/2) is discrete.
Since
I V2 | = | 2IV2,
we see that e = 2, and, since n = efyf — 1. Also,
1 > I V2| > 12 |e.
If we now form Q2(-v/2, ^2),then again wehaveaunique extension
of the valuation with e = 2, f = 1, and
1 > | ^2 | > | V2 | > | 2 |8.
We continue in this fashion and form
k = &(V2, ^2, #2,
Since k is algebraic over Q2, the valuation can be extended
uniquely to k, but it is clearly no longer discrete. We now form
the completion к, and let К = к(у/ — 1). We claim that in this
case the degree n of К over к is 2, while e = f = 1. Hence,
n =/= ef even though к is complete. First, we have that n = 2,
for у/— 1 ф Qt, and, therefore, у/— l ф k. It also does not
belong to к, for if д/— 1 =	 where the anek, then we
must have | an | = 1 for all n sufficiently large, since | д/— 1 | = 1.
Thus, д/ — 1 would be a limit of units where these units cannot
all belong to a finite dimensioned subfield of k since any such
field is complete, and, therefore, closed. This, together with our
earlier remarks on the various residue class degrees enables us to
assume that
“n = 1 + 5/2 Д, ,
where /?„ is also a unit, so
1 + Л/2^В-V^T,
150
V. Extensions of Valuations
which is clearly impossible. Hence the degree of К over k is 2.
Furthermore,
I yCTT _ 1 I = I 2 |v« < 1,
so 1 — 1) — 0, where <p is the associated place, and
1) = <p(—1). Hence f = 1. Finally, for a e K,
a = a + b — j, where a, b e k, and
| ® | = | aa + 6s p'«e | k |,
so e — 1.
Although e/ = n is no longer true if | | is not discrete even if k
is complete, it can be shown that in the complete case ef always
divides n. We shall not be able to go into these matters here.
The reader may consult Artin [1] for further considerations
along these lines.
We observed earlier in (2.5) of this chapter, in the case
of a finite separable extension, К = A(a) of degree n of k,
that n = nt, where the nt are the local degrees. Let
A(a) ~ k(a') C £(<*') be an embedding of A(a), where £(a') is of
degree п, over 4, and a' a root of some />,(x) over ft where the
pi(x) are the irreducible factors over 4 of the irreducible poly-
nomial satisfied by a over k. Since the residue class fields and
value groups of k and A are the same, and since they are also the
same for fe(a') and we have, in view of the previous theorem,
that, if the valuation | | on k is discrete, then
«= 2 *
r-1
where the ef and /, are the ramification indices and residue class
degrees of the various extensions of | | to K. We state this as a
theorem.
Theorem 4.3	. If К is a finite separable extension of degree
n of k, and if | | is a discrete valuation of k with r extensions to К
4. Discrete Valuations
151
which have ramification indices ex............. and residue class
degrees, respectively, then
я = X •
Let us suppose now that К is a finite separable normal extension
of degree n of k and that | | is a discrete valuation of k. Then, if
G is the Galois group of К over k, we have, in the notation of
Theorem 2.4 of this chapter, that G = Svt (disjoint),
where r is the number of extensions of | | to K. The local degrees,
nt, are the degrees of the vt{K} к over к, but v£K) = K, and,
therefore,
ei = ег = •" = e, — e,
However, by the previous theorem, n — SJLj whence,
n = efr. Thus, we have shown:
Theorem 4.4	. If К is a finite separable normal extension of
degree n of k, and if | | is a discrete valuation of k with r extensions
to K, then all the ramification indices are equal to the same
number e, and all the residue class degrees are equal to the same
number f. Moreover, n = efr.
As a final consideration, we shall prove an extended version
of the Eisenstein irreducibility criteria.
Theorem 4.5	. Let f(x) = xm +	+ ••• + atx + a0
be a polynomial over a field k with a discrete valuation | |. If
| тг | is the generator of | k* | with | n-1 <1, and if ir | a,, i = 0,
1,..., m — 1, but тг® 4" a0 , then f(x) is irreducible.
Proof, Let a be any root of/(x), and consider К — k(a). Let
я be the degree of К over k. Clearly, n m. Assume that the
valuation has been extended to К in some fashion and denote the
extended valuation by | | also. Now,
+ ат-р™-1 +	+<*!« + e0 = 0
152
V. Extensions of Valuations
which implies that | a | < 1, and, since we know that it | at,
i = 0,1, ...,m — 1 and ?r2 f a0, we have that
I a™-!»”'11....I I
are all less than | a0 |. However, there must be at least two
dominant terms in the equation, whence
I I = I I =4 l>
or | a | = 177 |1Zm. Thus, the ramification index, e, of the extension
is greater than are equal m. Hence,
m n e J? m.
Therefore, m — n — e, the residue class degree of the extension
must be 1, It is now also clear that f is irreducible, and the proof
has been completed.
A special case of this situation was observed in Example 5 of the
previous section. Extension fields, K, of k of degree n for which
e = n are called fully ramified.
Further extensions of Eisenstein’s criteria are possible, where
the valuation is no longer assumed discrete, but we refer the
reader to the references for these considerations.
This concludes our discussion of matters centering around the
extensions of valuations. Many important matters, such as the
ramification theory of valuations, we have barely touched on or
have not gone into at all. The interested reader is strongly
recommended to consult the books and notes listed in the
Bibliography, in particular, Artin [1], Bourbaki [5], Zariski and
Samuel [23], and Schilling [19].
1. Sets and Mappings
153
Appendix
1. Sets and Mappings
Let A be a set. If an element a belongs to A, we write a e A. If
a is not an element of A, we write аф A. Let {Аг} be a collection
of sets where it is understood that a runs over some index set
Л. We denote the union of the sets by Ua Ал. This is the set of
all elements which belong to at least one of the Ал. We denote
the intersection of the sets by Da Aa. This is the set consisting
of those elements which belong to all the Aa .
Let A and В be two sets, the difference set (or the complement of
В in Л) is denoted by A — B, and consists of those elements in
A which are not in B.
If the set A is contained the set B, i.e., if every element of A
is an element of B, we say that A is a subset of В and write
A С B, or В Э А. If А С В and В C A we define the sets to be
equal and write A = B.
The set consiting of no elements at all will be denoted by и and
is called the empty set or null set. We shall frequently use the
notation for a set:
{xeS|P(x)}.
This designates the set of all elements belonging to S which
satisfy a certain condition or proposition, designated by P(x).
If x e S, the set consisting of just x will be denoted by {x}.
If A and В are two sets, the cartesian product A X B, is the set
of all pairs (a, b) where a G A and beB.
Let A and В be two sets. If to each element a g A a unique
element b G В is associated, we say that there is a mapping or
function f from A into В and write f(a) = b. We denote this by
/: A -+B, or АЛВ.
If a± ф аг implies /(a^ Ф f(at), then f is said to be one-to-one.
If for each b e B, there exists an element a e A such that/(«) = b,
then /is called an onto mapping.
154
Appendix
Let / be a mapping of A into B. Then, if E C A, the image set
f(E) is the set {/(x) j x e £}. Let F С B, then the pre-image set
/~4F) = {x|/(x)eF}.
Next, if/: A —► В and if E C A, the restriction off to E is the
mapping, denoted by f , and given by f |£(x) = /(x) for all
xe E.
Finally, if
А Л В Л- C,
then the composite map (or product map) gf maps A into C and is
defined by {gf> (x) = g{f(x)).
Let S be a set and let ~ designate a relation defined between
pairs of elements of S such that, given any two elements a, b e S,
either a ~ b (read a is equivalent to b) is true or false. The relation
is called an equivalence relation if it satisfies the following con-
ditions:
(1)	a ~ a for all a e S (reflexivity);
(2)	a ~ ft implies ft ~ a (symmetry);
(3)	a ~ ft and ft ~ c implies a ~ c (transitivity).
A special equivalence relation is given by taking S = Z, the
set of all integers, and defining a ~ ft if and only if mja — ft
(i.e., if and only if m divides a — ft) where m is a fixed positive
integer. This special equivalence relation is denoted by a = ft
(mod m), read a is congruent to ft modulo m.
Suppose that 5 is a set and ~ an equivalence relation defined
on S. We denote by a, the set of all elements of 5 equivalent to a.
Such a set is called an equivalence class.
Theorem 1.1. If S is a set with an equivalence relation defined
between pairs of elements of S, then S is decomposed into disjoint
equivalence classes. We denote this by S — \J a. Here it is
understood that the union is taken over only certain a 6 S, so that
the sets are disjoint.
Now, let g: E -+F. We define, for a, ft e E, a ~ ft if and only
if g(a) = g(b). It is clear that this is an equivalence relation on E.
2. Number Theory
155
Let E denote the set of all equivalence classes and consider the
following mappings

where /(a) = а, Л(а) = g(a), and i(g(a)) — g(a). f is an onto
mapping, h is well defined and is one-to-one and onto, i is a
one-to-one mapping called the injection map. Finally, g = ihf.
A set S is said to be partially ordered if there is a relation,
denoted by <, defined between some pairs of elements a, b of 5
such that
(1)	a < b and b < c => a < c\
(2)	a < a for all a e S;
(3)	a < b and b < a => a = b.
If S is partially ordered, and, if for every pair a, b e S, a < b
or b < a, then 5 is called totally ordered.
Let S be a partially ordered set and let E C S. An element
b 6 S is called an upper bound of E if a < b for all a e E. If c 6 S
and if whenever aeS, c < a c = a, then c is said to be a
maximal element of 5.
A set S is called inductively ordered if 5 is partially ordered and
any totally ordered subset has an upper bound in S.
The following set theoretical axiom, equivalent to the axiom
of choice and many other set theoretical statements, will be
assumed.
Zorn’s Lemma. A non-empty inductively ordered set 5 has a
maximal element.
Applications of this axiom appear throughout the text.
2.	Number Theory
As usual, we denote by Z the set of all integers. If an integer a
divides an integer b, we denote this by a/b.
The greatest common divisor (g.c.d.) of two integers a and b is a
156
Appendix
positive integer d, denoted by (a, 6), such that d/а and d/b, and,
if c is any integer such that c/a and c/b, then cjd.
Theorem 2.1	. The g.c.d., d = (a, b), of any two integers a
and b exists, is unique, and can be expressed in the form
d = xa + yb, where x and у are integers.
If (a, b) = 1, then a and b are called relatively prime. The
least common multiple (l.c.m.) of two integers a and b is a positive
integer -r, denoted by {a, b}, such that а/т and b/т, and, if afc and
bjc, then т/с.
The l.c.m. of any two integers a and b exists and is unique.
In Section 1 of the appendix, we introduced the equivalence
relation a = b (mod m) in Z. The equivalence classes for this
special equivalence relation are called residue classes (modulo m),
and a set of elements, one from each class, is called a complete
residue system (modulo m).
We denote by ^(m), the Euler ^-function, the number of
positive integers less than or equal to m and relatively prime to m.
If n =nw^jl, where the/), are distinct primes, then
Theorem 2.2	. (Euler’s theorem). If (a, m) = 1, then a*<m) == 1
(mod m).
Theorem 2.3	. (Fermat’s theorem). If p is a prime and if
(a, />) = 1, then в”-1 = I (mod/>).
Theorem 2.4	. The linear congruence ax == b (mod m) has
a solution if and only if d]b, where d = (a, m). If d]b, then the
congruence has d solutions modulo m.
Theorem 2.5	. (Chinese remainder theorem). Given the
system of congruences
x = (mod mJ
x = a2 (mod mJ
x ~ an (mod mn)
2. Number Theory
157
where (mf, m}) — 1 for i Ф j. There exists a unique solution of
the system modulo
mtma ••• mn.
Theorem 2.6	. Let A denote a complete residue system
modulo a prime p. Then for any a 6 Z,
a s a0 + atf> + ••• + On-ip""1 (modpn)
where the a( e A. This representation is unique.
Consider now the quadratic congruence
X2 * * s * = a (mod p)
where p is a prime. An integer a, where (a, p) = 1, is called a
quadratic residue modulo p if this equation has a solution. If a is
not a quadratic residue modulo p, it is called a quadratic nonresidue
modulo p.
If p is an odd prime, the number of quadratic residues modulo
p is (p — l)/2, and, consequently, the number of quadratic non-
residues modulo p is also (/> — 1 )/2.
For (a, p) = 1, the Legendre symbol, (ajp), is defined as follows:
©i 1 if a is a quadratic residue mod p
|—1 if a is a quadratic nonresidue mod p
We list the following properties of (alp):
(2) a = b (mod p), then (-
(7)=(>
(4)
(5)
(6)
(mod/») if p is an odd prime.
(—-) = (—i>/a if p is an odd prime.
' P >
c j j •	* A (1 ifp = ±1 (mod 8)
For an odd prime p, (-) - jf# ±J (m0<J 8)
Finally, one has:
158
Appendix
Theorem 2.7	(quadratic reciprocity law). If p and q are odd
primes, then

!)/«].
For proofs of the preceding statements, one may consult
Niven and Zuckerman [16], Hardy and Wright [8], LeVeque [13].
3.	Groups
Definition 3.1. A group is a set G together with an operation
defined between pairs of elements a, beG (we denote the
operation by a • b, or simply ab, and call it multiplication) which
satisfies the following axioms:
(1)	For all a, b e G, ab 6 G (closure).
(2)	a(bc) = (aft) c (associativity).
(3)	There exists an element 1 e G (called the identity element)
such that a • 1 = 1 • a = a for all a e G.
(4)	To each element a e G, there exists an element a1 e G
(called the inverse element to a) such that aa~x = a~la = 1.
A set G satisfying just the first two axioms is called a semigroup.
If G is a group and if for all a, ft 6 G, ab = ba, then G is called
commutative or abelian. In this case, the operation is usually
denoted by +, and one writes a + ft instead of ab. The identity
is then written as 0, and the inverse of a as —a. Finally, one
writes a — ft instead of a + (—ft).
Definition 3.2. A group G which contains only a finite number
of elements is called a finite group. The order of such a group,
denoted by ord G, is the number of elements of G.
Definition 3.3. A subset H of a group G is called a subgroup if
(1)	a, ft e H implies ab e H;
(2)	16Я;
(3)	aeH implies a-1 eH.
3. Groups
159
It follows from the associativity axiom for a group G that the
associative law holds for any finite number of elements, and, if
, aa ,.... a* belong to G, we can write unambiguously, агаа
If all the at = a, we write a”, or na in the additive case. If for
some positive integer n, an = 1, then a is said to have finite order,
and the smallest such positive integer n is called the order of a
and is denoted by ord a.
Theorem 3.1	. If a has finite order, and if ak = 1, then
ord a | k.
Let G be a group and H a subgroup of G. Define a ~ b if and
only if tr'beH. It is readily seen that this is an equivalence
relation and that a ~ b if and only if b e aH. Thus, the equi-
valence classes are the sets of the form aH, called left cosets
of H. We have that either aH = bH, or aH and bH are disjoint.
Furthermore,
G = U aH,
where the union is disjoint and taken over distinct left cosets.
One could proceed analogously be defining a ~ b if and only
if ba-1 e H; one then gets a decomposition of G into right cosets,
Ha.
Theorem 3.2	. (Lagrange). The order of a subgroup H of a
finite group G divides the order of G.
We write [G : H] — and call [G : H] the index of H
in G.	Г
Theorem 3.3	. The order of any element of a finite group G
divides the order of G, and if ord G = я, then an = 1 for all
ae G.
Suppose there exists an element a belonging to the group G
such that every b G G is of the form an, then G is called a cyclic
group and a is said to be a generator of G. We write G = [a].
If and Sa are subsets of a group the product is the set of
all elements of the form aaaa where aa g Sx and a2 G S2 . The
160
Appendix
associative law in G implies that S/SgSg) = (SXS2) S2, where
Sx, S2, S3 are three subsets of G. If Sx consists of just a single
element a, we write aSt instead of SXS2 . This is in agreement with
our previous notation for left cosets. Clearly, if Я is a subgroup of
G, then Я2 = H.
Definition 3.4. A subgroup N of G is called normal if
aNa-1 = N for all a e G.
In the case of an abelian group, it is clear that every subgroup
is normal.
If N is a normal subgroup of G, and if aN and bN are any two
left cosets, then
(aN) (bN) = abN2 = abN.
Motivated by this, we consider the set G/N of all left cosets
aN, aeG, where N is a normal subgroup of G. It can be shown
that the set G/N is a group, called the factor or quotient group,
with respect to the operation of set multiplication, i.e.,
aN  bN = (aN) (bN) = abN.
If G is a finite group, then ord (G/N) =	= [G : NJ.
Definition 3.5. Let G and G' be two groups, and let /:
G —G'. f is called a homomorphism of G into G' if
f(ab) = f(a)f(b) for all a, b G G. If f is also onto, then Gr is called
a homomorphic image of G. If f is a one-to-one homomorphism, it
is called an isomorphism, and, if it is onto G', G' is called an
isomorphic image of G, and we write G' ~ G. An isomorphism
of a group onto itself is called an automorphism.
An improtant example of a homomorphism is the following.
Let G be a group and N a normal subgroup; map G —► G/N by
the mapping a —aN for all a G G. This mapping is a homo-
morphism of G onto G/N and is called the canonical homo-
morphism.
4. Rings, Ideals, and Fields	161
Iff: G -* G' is a homomorphism, the set
К = {aeG\f{a) = 1'},
where Г is the identity of G' is readily seen to form a normal
subgroup of G and is called the kernel of the homomorphism.
Theorem 3.4	(fundamental theorem of homomorphisms). If
f: G -* G' is a homomorphism of G onto G', then G' ~ G/K,
where К is the kernel of f.
Again, let G be a group and let HY and Ht be two subgroups of
G. Define for a, b 6 G, a ~ b if and only if hiaht — b, where
1^ e Hi and hse Нг. This is easily seen to be an equivalence
relation. The equivalence classes are sets of the form and
are called double cosets with respect to Нг and Нг . We have
G = U НхаН* (disjoint)
where the union is taken over certain elements a e G. If G is finite,
unlike the case of cosets, distinct double cosets need not contain
the same number of elements. One can show that the number of
elements in НгаН2 is given by
ord ord Hi
ord (Яг n a-1Kxa)
The following references are suggested for further considerations:
Hall [7], Ledermann [12], Zassenhaus [24], Kurosh [11].
4.	Rings, Ideals, and Fields
Definition 4.1. A set A together with two operations, denoted
by + and •, defined between pairs of elements a, b e A is called a
commutative ring if:
(I)	A, with respect to the operation +, is an abelian group;
(2)	A, with respect to the operation •, is a semigroup;
(3)	ab — ba for all a, b 6 A;
(4)	a(b + c) — ab + ac for all a, b, c in A (distributive law).
162
Appendix
We shall in the future omit the word commutative, it being
understood that all rings we consider are commutative.
A ring A is called a ring with identity if there exists an element
1 6 A such that a • 1 = a for all a E A.
It follows immediately from item 4 in the definition of a ring
that a • 0 = 0 for all a e A.
Definition 4.2. If A is a ring and if ab = 0 implies a = 0 or
6=0, then A is called an integral domain.
Definition 4.3. If A is a ring and if A* = A — {0} is an
abelian group with respect to •, then A is called afield.
An element a belonging to a ring A is called a unit if a has a
multiplicative inverse in A.
Definition 4.4. If A is a ring, a subset I is called an ideal if:
(1) a, 6 6 Z implies a — 6 e 7;
(2) a 6 A, bel implies abel.
If a el, an ideal, we frequently write a = 0 (mod/).
With respect to the operation +, condition 1 states that an
ideal, I, is a subgroup of A, and, since A is an abelian group
with respect to +, I is a normal subgroup, and we can consider
the factor group A [I of all cosets a + I where a e A. A/Ican be
considered as a ring, called the quotient ring, by defining
(ai + Z) • {at +1) = a^z + Z,
where , a2 e A.
Definition 4.5. Let A and A' be two rings. A map/: A —* A'
is called a homomorphism if, for all a, b e A,f(a -j- 6) = f(a) 4-/(6),
and/(d6) = /(a)/(6). If/is also one-to-one, it is called an isomor-
phism, and should / also be onto, A' is called an isomorphic
image of A. We designate this by A ~ A'. An isomorphism of a
ring onto itself is called an automorphism.
Let/: A —>- A' be a homomorphism. The set
A = {aGJ|/(*) = 0'}
4. Rings, Ideals, and Fields
163
where 0' is the additive identity of A', is easily seen to be an
ideal of A and is called the kernel of the homomorphism.
Analogous to the results on groups in the previous section, we
have that the map
f’A-*A/I,
where 1 is an ideal of A, and where /(a) = a 4-1, is a homo-
morphism, which we call the canonical homomorphism. Also,
if^ : A —> A' is a homomorphism of A onto A', then A' ~ A/K,
where К is the kernel.
If A is a ring with identity, then the set
(a) = aA = {ab | b e A}
is an ideal containing a and is called the principal ideal generated
by a. More generally, if S is a subset of A, where A has an
identity, then the set
(S) =
where we consider only finite sums, is an ideal containing 5 and is
called the ideal generated by 5.
For the ring Z of integers, we have the following result.
Theorem 4.1	. In the ring Z, every ideal is a principal ideal.
Another example of such a ring is the ring F[x], of all poly-
nomials in a transcendental element x over F with coefficients
from the field F.
Definition 4.6. An ideal I in a ring A is called a prime ideal if
A[I is an integral domain. An ideal I in a ring A is called maximal
ifl^A and if J is an ideal such that J Э / (proper inclusion, i.e.,
J ф I), then J — A.
One can show that given any ideal ^A, where A is a ring with
identity, there exists a maximal ideal containing it.
164
Appendix
Theorem 4.2	. If A is a ring with identity and if M is a maximal
ideal in A, then M is a prime ideal. Moreover, AfM is a field.
Conversely, if А/M is a field, then M is a maximal ideal.
It is easy to see that a field, F, has no ideals other than F itself
and {0}. This result implies that the only homomorphic images of
a field are the trivial ones, where we map all elements into the
additive identity, or are isomorphic images.
Let F be a field. Suppose па = 0 for some a e F. Then, if b is
any other element ofF, nb = 0. One says thatF has characteristic
p if there exists a positive integer/» such that pa = 0 for all a e F,
where p is the smallest positive integer with this property. If F
does not have characteristic p, one says that F has characteristic 0.
If F has characteristic />, then it is easy to see that p must be a
prime.
Again, suppose that F is a field. The intersection of all subfields
of F is clearly a subfield, Fo , of F and is called the prime field of F.
One can prove that the prime field Fo of any field F is either
isomorphic to Q or to Zfip) for some prime/».
5. Glossary for Rings and Fields
In this section, we shall just list some definitions concerning
rings and fields for handy reference for the reader.
Rings
1.	Let A be a commutative ring with identity. If a, be A,
then b 0 divides a, written 6/a, if there exists an element ce A
such that a — be. If a/b, and b/a, a and b are called associates.
An element a e A is called a prime (or an irreducible element) if
bja implies that b is a unit of A or b is an associate of a.
2.	Let A be a ring contained in a field F, and S # 0, a subset
of A such that 0$S, The set
I a e A, b e
(о I	1
is a ring, called the quotient ring of A by S, which contains A and
in which all elements of S have inverses.
5. Glossary for Rings and Fields
165
3.	If A is a (commutative) integral domain with identity in
which every ideal is principal, A is called & principal ideal domain.
4.	A (commutative) integral domain with identity in which
every non-unit and non-zero element can be written uniquely as
a product of primes except for order and multiplication by units
is called a unique factorization domain.
5.	Let A be a (commutative) integral domain with identity.
Suppose there exists a function 8 : A —► Z such that for a, b E A
(a)	8(a) > 0 and = 0 if and only if a == 0;
(b)	8(o6) = 8(a) 8(6);
(c)	if b Ф 0, there exist elements q, r in A such that a = bq 4- r,
where 8(r) < 8(6).
Then A is called a euclidean domain.
Fields
1.	If К is a field and if k is a subfield of K, then К is called an
extension field of k.
2.	Let К be an extension field of k. Then К can be viewed as a
vector space over k. If К is finite-dimensional over k, then К is
called a finite extension of k. The dimension of К over k is also
called the degree of К over k.
3.	If К is an extension field of k, and if a e К satisfies a poly-
nomial equation f(x) = 0 with coefficients in k, then a is called
algebraic over k\ otherwise, a is called transcendental over k. The
extension К is called algebraic over k if every element of К is
algebraic over k.
4.	An extension field К of k is called normal if it is algebraic
over k and if every irreducible polynomial in fc[x] which has a root
in K, factors into linear factors in K[x].
5.	Let « e K, an extension field of k. If a satisfies an irreducible
polynomialf(x) 6 6[x], then a is called inseparableover k if/'(“)= 0;
otherwise, a is called separable over k. An algebraic extension
field К of k is called separable if every a E К is separable over A;
otherwise, К is called an inseparable extension of k.
6.	Let К be an extension field of a field k of characteristic p.
166
Appendix
An element a. e К is called purely inseparable over k if a”* e k for
some integer e 0. If every element of К is purely inseparable
over k, then К is called л purely inseparable extension of k,
7.	If К is a finite extension field of k, the set of all elements of
К algebraic over k form a subfield, Кг . The degree of К over
Kx is a power, pf of the characteristic p of k and is called the
degree of inseparability of К over k. The degree of over k is
called the degree of separability of К over k.
8.	If К is an extension field of k, and if S is a subset of K, then
the intersection of all subfields of К which contain S and k is a
field, called the field generated by S in К over k.
9.	Let /(x) be a polynomial with coefficients in a field k. A
splitting field of f(x) over k is an extension field К such that f(x)
decomposes into linear factors inA[x] and such that К is generated
over k by the roots of f(x) in K.
10.	If К is an extension field of k, and if a G К and a algebraic
over k implies a e k, then k is called algebraically closed in K.
11.	If К is an extension field of k, then К is called an algebraic
closure of k provided К is an algebraic extension of k, and К is
algebraically closed, i.e., К has no proper algebraic extensions.
12.	If К is a finite normal extension of k, the group of auto-
morphisms of К which leave k fixed (elementwise) is called the
Galois group of К over k.
13.	Let К be a finite separable extension field of k and let E
be the least separable normal extension field of k containing K.
If G is the Galois group of E over k and if H is the subgroup of G
leaving К fixed, write G = U,- (disjoint). Define for a G K,
the norm, NK/k(oi), from К to k of a and the trace, SKlk(a), as
follows:
Wjr. (“) = П
I*	4
5K| («) = X
I*	i
6. Addles and Iddes
167
If К is a finite inseparable extension of k, let pf be its degree of
inseparability. Then define
ЛГг, («) = ГЫ*)".
I* i-1
where the run over the distinct isomorphisms of К (leaving k
fixed) into the least normal extension of k containing K.
For references see Van der Waerden [21], Zariski and Samuel
[23], Jacobson [10], and Artin [2].
6. Adiles and Idiles
This final section is not a review section as the preceding
ones. We just want to define some notions here in a rather
special setting. We shall not go beyond a few definitions. How-
ever, because of the importance of these notions in such areas as
class field theory and algebraic geometry, we feel that some intro-
ductory mention of them should be made. For a more general
framework, as well as for applications, one may consult Artin [1],
Artin-Tate [4], Weil [22], and Chevalley [6].
We shall denote by S = {| |„} a set containing all inequivalent
/>-adic valuations as well as the usual absolute value, which we
denote here by |	, p = We now form the cartesian product
°f the sets Q„ , where denotes R. Elements of this
product are of the form
x = (..., ap , ...)
where a9 6	. This set can be made into a ring by adding two
such elements componentwise and by multiplying them com-
ponentwise also. The field Q itself can be mapped isomorphically
into this ring by the mapping
a —► (a, a, ...)	(6.1)
where a 6 Q.
168
Appendix
We now consider the subring AQ consisting of those x with
| av | „ < 1 for all but a finite number of p. AQ is clearly a subring
which is called the adele ring of Q. The elements of AQ are called
adeles or valuation vectors of Q. Certainly, for any a G Q, | a |P 1
for all but a finite number of p, so AQ contains the isomorphic
image of Q under (6.1).
Next, we define for any x G ПР QP
I x I» = I av Ip •
Thus, x G Aq if and only if | x |p < 1 for all but a finite number
of p.
We consider the set I of all x e AQ such that | x | „ Ф 0 for all p,
and | x |0 = 1 for all but a finite number of p. 1 is easily seen to
form a multiplicative group, and the elements of I are called
ideles oiQ. If a gQ*, then | u 0 for any prime and | a |„ = 1
for all but a finite number of p, so I contains an isomorphic
image of Q* under the map (6.1).
We finally note that a topology may be introduced in AQ by
taking a neighborhood basis of 0 G Ao to consist of all sets
Px ={убЛ0 | |y|p < | *(„}
where x el.
Bibliography
1.	Aktin, E., Algebraic Numbers and Functions (lecture notes),
Princeton, 1951.
2.	Artin, E., Galois Theory, Notre Dame, 1948.
3.	Artin, E., Elements of Algebraic Geometry (lecture notes), New
York University, 1955.
4.	Artin, E., and Tate, J., Class Field Theory (lecturenotes),Princeton,
1951-1952.
5.	Bourbaki, N., “Algebre Commutative.” Hermann, Paris, 1961.
6.	Chevalley, C., Class Field Theory, Nagoya University, 1953-1954.
7. Hall, M., “The Theory of Groups.” Macmillan, New York, 1959.
8.	Hardy, G., and Wright, E., “An Introduction to the Theory of
Numbers.” Oxford Univ. Press, London and New York, 1954.
9.	Н ille, E., andPHiLLips, R., “Functional Analysis and Semi-Groups,”
Amer. Math. Soc. Colloquium Publications, 1957.
Bibliography
169
10.	Jacobson, N., “Lectures in Abstract Algebra,"Vol. 1. Van Nostrand,
Princeton, New Jersey, 1951.
11.	Kurosh, A., “The Theory of Groups” (2 volumes). Chelsea, New
York, 1955.
12.	Ledermann, W., “The Theory of Finite Groups.” Oliver & Boyd,
Edinburgh and London, 1953.
13.	LeVeque, W., “Topics in Number Theory” (2 volumes). Addison-
Wesley, Reading, Massachusetts, 1956.
14.	Lorch, E., “Spectral Theory.” Oxford Univ. Press, London and
New York, 1962.
15.	Naimark, M., “Normed Rings,” Noordhoff, Groningen, 1959.
16.	Niven, I., and Zuckerman, H., “An Introduction to the Theory
of Numbers.” Wiley, New York, 1960.
17.	Ostrowski, A., Uber einige Losungen der Funktionalgleichung
$К*Жу) — <p(xy). Acta. math. 41 (1918), 271-284.
18.	Roquette, P., On the prolongation of valuations. Trans. Amer.
Math. Soc. 88 (1958), 42-57.
19.	Schilling, O., “The Theory of Valuations,” Mathematical Survey s,
American Mathematical Society, 1950.
20.	Tornheim, L., Normed fields over the real and complex numbers.
Michigan Math. J. 1 (1952), 61-69.
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1958.

Index
Absolute convergence, 98
Adeles, 167
Algebraic
closure, 166
integer, 89
Analyticity, 114
Approximation theorem, 21
Archimedian ordered group, 78
Associated place, 69
Associated valuation, 74
Associated valuation ring, 68, 73
Banach algebra, 111
Banach space, 94
Binomial series, 51
Bounded linear functional, 100
Canonical homomorphism, 160,
163
Cauchy sequence, 16, 94
Characteristic, 164
Chinese Remainder Theorem, 22,
156
Commutative normed algebra, 111
Complete
field, 24
residue system, 156
Completion of a field, 26-33
Congruence, 26
Conjugate embeddings, 130
Convergence
absolute, 98
p-adic valuation, 4
rank one valuation, 24
Convex functional, 103
Degree
extension, 165
of separability, 166
Discrete valuation, 143
Discriminant, 142
Divisibility, 164
Domain of convergence, 43
Double coset, 161
Eisenstein criteria, 151
Equivalent
norms, 94
places, 70
valuations, 16
Euclidean domain, 165
Euler
9>-function, 156
theorem, 156
Exponential series, 46, 47
Extension
algebraic, 165
field, 165
finite, 165
normal, 165
separable, 165
theorem, 83-88
transcendental, 165
Extension of a valuation
archimedian, 126-128
non-archimedian, 117-126
Fermat’s theorem, 156
Fully ramified extension, 152
Gelfand theorem, 116
Global
degree, 132-133
norm, 136
trace, 136
Greatest common divisor (G.C.D.),
155-156
Group
Galois, 166
ordered, 70
Group of units, 23, 67
171
172
Index
Hahn-Banach theorem, 103, 106,
107
Ideal, 163
Idfeles, 168
Identity element, 111
Induced valuation, 130
Inductively ordered, 155
Integral
closure, 89
element, 88
Irreducible element, 164
Isolated subgroup, 77
Isometry, 26
Kernel, 161
Lagrange’s theorem, 159
Least common multiple, 156
Legendre symbol, 157
Liouville’s theorem, 114
Linear functional, 99, 100
norm of, 101
Local
degree, 133
norm, 136
trace, 136
Logarithmic series, 48, 49
Metric, 3, 4, 15, 93
Metric spaces, 3, 4, 15, 93
Neighborhood (spherical), 15
Newton’s method, 52-57
Non-units, 65
Norm, 92, 166
Normalized p-adic valuation, 23
Normed algebra, 111
Normed linear space, 92
Null sequence, 16
Order isomorphism, 78
Ordered group, 70, 77
Ordinal, 45
p-adic
integer, 37
numbers, 33, 34
numbers (canonical expansion
of), 35
valuation, 2
Place, 67-70
Power series in Qp , 43
Prime, 164
Prime field, 164
Principal ideal domain, 165
Quadratic
non-residue, 157
reciprocity law, 158
residue, 157
Radius of convergence (power
series), 43, 98
Ramification index, 121
Rank
of an ordered group, 77
of a valuation, 77
Regular point, 113
Relatively prime, 156
Residue
class degree, 121
class field, 9, 67
classes, 156
Roots of unity in Qt, 61
Semigroup, 158
Spectrum, 113
Splitting field, 166
Subgroup, 158
normal, 160
Sublinear functional, 103
Symmetric convex functional, 107
Trace, 167
Trivial valuation ring, 65
Ultra-metric inequality, 3, 4
Unique factorization domain, 165
Index	173
Unit, 162
Units, Group of, 23, 67
Valuation
associated, 74
discrete, 143
equivalent, 16, 75
general, 72
induced by an embedding, 130
non-archimedian, 5, 72
normalized p-adic, 23
of algebraic number field, 137
of rank one, 5
p-adic, 2, 5
ring, 9, 65
trivial, 9
vector, 168
Zorn’s lemma, 155