The Theory of Numbers 0720424585, 9780720424584

Table of contents :
Preface
Contents
Table of Notations
CHAPTER I. Cohomology of Groups
§1. Tensor products and groups of homomorphisms
1.1. g-modules
1.2. Tensor products
1.3. Groups of homomorphisms
§2. Homology and cohomology
2.1. Homology of complexes
2.2. Coefficient modules
2.3. Homology and cohomology of groups
2.4. The Duality Theorem
§3. Inhomogeneous complexes \mathscr{C}( \mathfrak{g} )
3.1. Definitions
3.2. The first homology and cohomology groups
3.3. Extensions of groups and the second cohomology groups
§4. Subgroups and related cohomology groups
4.1. Homomorphisms of groups and their cohomology theory
4.2. Cohomology groups of conjugate subgroups
4.3. Restriction and transfer
§5. Tensor products and cup products
5.1. Direct products of groups and their cohomology theory
5.2. Cup products
5.3. Cup products and other mappings
§6. Cohomology theory of finite groups
6.1. Dual modules
6.2. Cohomology groups of negative degrees
6.3. Restriction and transfer
6.4. Cup products
§7. Cohomology theory of cyclic groups
7.1. Monomial complexes
7.2. The Herbrand quotient
§8. Tate's Theorem and Galois cohomology
8.1. Tate's Theorem
8.2. Galois cohomology
CHAPTER II. Valuation Theory
§1. Valuations of fields
1.1. Valuations
1.2. Non-Arcbimedean valuations and Archimedean valuations
1.3. The Approximation Theorem and the Independence Theorem
1.4. Valuations of the rational number field and of a rational function field
§2. Complete fields
2.1. Completions
2.2. Remarks concerning normed spaces
2.3. Extensions of a valuation of a complete field
§3. Archimedean valuations
3.1. The determination of complete Archimedean valuation fields
3.2. Non-complete cases
§4. Non-Archimedean valuations I
4.1. Valuation rings, valuation ideals and residue class fields
4.2. Hensel's Lemma
4.3. Extensions of a valuation
4.4. Discrete valuations
4.5. The system of multiplicative representatives
§5. Non-Archimedean valuations II
5.1. Tensor products of vector spaces
5.2. Compositions of fields
5.3. Extensions of valuations
5.4. Non-Archimedean valuations of an algebraic number field
5.5. Valuations of an algebraic function field
§6. Hilbert's theory
6.1. Decomposition groups and inertia groups
6.2. The decomposition of prime spots in intermediate fields
6.3. Frobenius substitutions
6.4. Ramification groups
6.5. The ramification of a prime spot in intermediate fields
§7. Discriminants and differents (local cases)
7.1. Discriminants
7.2. Differents
§8. The differential of formal power series
CHAPTER III. Adele Rings and Idele Groups
§1. Locally compact groups
1.1. Haar measures
1.2. Duality
1.3. Fourier transformation
§2. Locally compact rings
2.1. The multiplicative topology
2.2. Norms of invertible elements
2.3. The topology of localJy compact fields
2.4. The classification of locally compact fields
§3. Local fields
3.1. Local fields
3.2. Self-duality
3.3. The multiplicative group k_p^*
§4. Adele and idele
4.1. The restricted direct product
4.2. Character groups
4.3. Adele rings
4.4. ldele groups
§5. Extensions of the base field
5.1. Isomorphisms of base fields
5.2. Extensions of base fields
5.3. Trace and norm
§6. The structure of adele rings
6.1. Expansion in partial fractions
6.2. The structure of adele rings
6.3. Self-duality
6.4. Normal characters
§7. The structure of idele groups
7.1. The structure of idele groups
7.2. The structure of idele class groups
CHAPTER IV. The Main Theorems of Class Field Theory
§1. Cyclotomic fields
1.1. General cyclotomic fields
1.2. Cyclotomic fields over the rational number field
1.3. Differents and discriminants of cyclotomic fields
§2. Kummer fields
2.1. Algebraic theory of Kummer fields
2.2. Examples
§3. Power residue symbols and Hilbert norm residue symbols
3.1. Definitions of the symbols
3.2. The self-duality of k_p^* / k_p^n (local cases)
3.3. The global cases (self-duality of J_k / J_k^n)
§4. Quadratic number fields and the Reciprocity Laws for quadratic residues
4.1. Calculation of (a, b/2)
4.2. Decomposition Law for quadratic number fields
4.3. The Reciprocity Laws for quadratic residues
§5. Artin-Schreier fields
5.1. Artin-Schreier fields
5.2. Artin-Schreier fields over k = F_q((t)) (Duality of k^* / k^P and k/ \mathscr{P}k )
5.3. Artin-Schreier fields over an algebraic function field (The duality of J_k / J_k^p and R_k/ \mathscr{P}R_k )
§6. The theory of infinite Galois extensions
6.1. The Galois theory
6.2. The maximum Abelian extensions and the character groups of their Galois groups
§7. Main theorems of the class field theory
7.1. Takagi groups and Artin correspondences
7.2. Chevalley's formulation
7.3. Ideal class groups, conductors
7.4. The order of proofs
CHAPTER V. Proofs of the Main Theorems
§1. Local cases
1.1. Cohomology groups of local fields
1.2. Invariants and canonical classes
1.3. Norm residue symbol
1.4. Relations with the Hilbert symbol
§2. Proofs of the Conductor Theorems
2.1. A theorem on the ramification groups
2.2. The Conductor Theorems
§3. The first inequality
3.1. An extension of Herbrand's Lemma
3.2. Calculation of Q(C_K)
§4. The Second Inequality and the Existence Theorem
4.1. Simplification of the problem
4.2. Proof of the Second Inequality (case (a))
4.3. Proof of the Existence Theorem (for case (a))
4.4. Proof for case (b)
§5. The Reciprocity Law
5.1. Cyclotomic extensions and the Law of Reciprocity
5.2. Idele class groups and their cohomology
5.3. Summation formula of p-invariants and product formula of the norm residue symbols
§6. Weil groups
6.1. Weil groups and transfer
6.2. The Weil groups; Local cases
6.3. Algebraic function fields over finite fields and their Weil groups
6.4. Weil groups of algebraic number fields
6.5. Decomposition groups and Inertia groups
APPENDIX 1. Ideal Theory
§1. Ideals in a Dedekind domain
1.1. Dedekind domains
1.2. Valuation theoretical characterization of a Dedekind domain
1.3. Extensions of a Dedekind domain
1.4. Extensions of ideals and their relative norms
§2. Discriminants and differents (Global Cases)
2.1. Differents
2.2. Discriminants
2.3. Miscellaneous theorems on differents and discriminants
2.4. The absolute discriminant and Minkowski's Theorem
2.5. An example
§3. Artin-Whaples' Theory
APPENDIX 2. History of the Class Field Theory
§1. From Euclid to Hilbert
1.1. The birth of Number Theory and the accomplishment attained by Gauss
1.2. Dirichlet and L-functions
1.3. Kummer and Dedekind
1.4. Complex multiplication
1.5. Hilbert's 'Zahlbericht' and Hilbert's problems
1.6. Hensel and p-adic numbers
§2. Takagi and Artin's class field theory
2.1. Weber's generalized ideal classes
2.2. Takagi's class field theory I
2.3. Takagi's class field theory II and Artin's Reciprocity Law
2.4. Hasse's Bericht
§3. The development of the theory after Takagi and Artin
3.1. Simplification of the proofs by Artin, Herbrand, and Chevalley
3.2. 'Arthmeticization' by Chevalley
3.3. The class field theory and the algebra theory
3.4. Hasse principle, idele and adele
3.5. Algebraic function fields over finite constant fields, algebraic geometry
3.6. The Weil groups, application of the cohomology theory and the class formation
3.7. Ideal class groups, class numbers and some miscellaneous facts
Bibliography
Index

Citation preview

THE THEORY OF NUMBERS

North-Holland Mathematical Library Board of Advisory Editors:

M. Artin, H. Bass, J. Eells, W. Feit, P. J. Freyd, F. W. Gehring, H. Halberstam, L. V. Hormander, M. Kac, J. H. B. Kemperman, H. A. Lauwerier, W. A. J. Luxemburg, F. P. Peterson, I. M. Singer and A. C. Zaanen

VOLUME 8

N·H

c1)~c

~

ffl

NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM • OXFORD AMERICAN ELSEVIER PUBLISHING COMPANY, INC. - NEW YORK

The Theo·ry of Numbers Edited by

S. IYANAGA Tokyo, Japan

Translated by K. Iyanaga

I

1975

NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM • OXFORD AMERICAN ELSEVIER PUBLISHING COMPANY, ING, - NEW YORK

NORTH-HOLLAND PUBLISHING COMPANY - 1975 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any farJJI or by any !11eans, electronic, mechanical, photocopying recording or otherwise, without the prior permission of the publisher.

Library of Congress Catalog Card Number: 75-23202 North-Holland ISBN for this Series: S O 7204 2450 X North-Holland ISBN for this Volume: 0 7204 2458 5 American Elsevier ISBN: 0 444 10678 2

A translation of; SURON by Shokichi Iyanaga ( ed.) Published by Iwanami Shoten, Publishers, Tokyo, 1969 Copyright © 1969 by Shokichi lyanaga

Publishers:

North-Holland Publishing Company - Amsterdam North-Holland Publishing Company, Ltd. - Oxford

Sole distributors for the USA and Canada:

American Elsevier Publishing Company, Inc. 52 Vanderbilt Avenue New York, N.Y. 10017 Library of Congress Cataloging in Publication Data

Iyanaga, Shokichi, 19o6The theory of numbers. (North-Holland mathematical library v. 8) Translation of Suren, Bibliography:17 p, Includes index. i. Class field theory. I. Title. Qf\247. r9613 1975 512'. 74 75-23202 ISBN 0-444-lo678-2

Printed in England

Preface The history of number theory has its origin in remote antiquity, yet at the same time it is very much alive today. Problems and discoveries of number theory stimulated the development of various other fields of mathematics, and, in turn, number theory itself progressed by adopting the new notions and methods supplied by other fields. For example, algebraic notions as fields, rings and ideals which are now widely used by mathematicians were originally introduced by Dedekind and others towards the end of the last century for the purpose of giving a systematic description of algebraic number theory. The emphasis on the importance of algebraic and topological methods, which led to the 'modernization' of twentieth century mathematics, naturally influenced number theory. Specifically, since the latter half of the 1940's, when cohomology theory (which has its origin in topology) was first amalgamated into algebra, number theory went through a radical change. Number theory is rich in problems which have remained unsolved for many centuries. A large part of the theory is still wrapped in mystery. It contains, on the other hand, magnificent theories such as class field theory. Class field theory was substantially completed by Takagi's paper published in 1920, and E. Artin's proof of the Reciprocity Law appeared in 1927. The subsequent discovery of new methods, however, simplified or changed some of the proofs and descriptions of the theory. The main purpose of this book is to give a systematic exposition of class field theory, with complete proofs, utilizing cohomology theory. Chapter I deals with cohomology theory, which is later utilized in the proofs of class field theory. As prerequisites for Chapter I we only assumed some basic knowledge of algebra, such as contained in Survey of Modern Mathematics I, Iwanami Shoten (in Japanese) or in van der Waerden's Algebra I, II, and knowledge of Galois theory. Chapter II deals with valuation theory. Also for this chapter we did not _assume any other prerequisites than the ones needed for Chapter I. Chapter III deals with adeles and ideles; the prerequisite knowledge concerning the locally compact topological groups is summarized at the beginning of the chapter (cf. the opening paragraph of the chapter for references). Chapters IV and V expound the class field theory. Chapter IV contains explanations of some specific fields such as cyclotomic and Kummer fields, which we believe, are essential for the understanding of class field theory. We adopted Chevalley's formulation of the main theorems of class field theory, namely, Theorems A, B, C in §7 of Chapter IV, which are stated V

vi

PREFACE

after an explanation of the classical Reciprocity Law. The proofs of the main theorems are given in Chapter V. To describe class field theory in the above mentioned chapters we utilize valuation theory and the notion of divisors. The classical notion of ideals, which appears more often in the traditional writings on number theory than its equivalent, the divisors, is treated in Appendix 1. Appendix 2 contains a. survey of the history of number theory, the history of the development of class field theory, and specifically, the recent history since the 1920's during which the theory gradually changed its appearance. The reader might find it helpful to refer to Appendix 2 first in order to acquire the historical perspective of class field theory. This volume is part of a series on 'Modern Mathematics' published by Iwanami Shoten, and the publication project dated from the early 1950's. The long delay in publication is due to circumstances for which the editor himself is solely and directly responsible. We first asked Professor Tannaka to compile this volume, and we received a graceful consent. To our great disappointment, however, heavy departmental duties forced him, in 1953, to withdraw from the compilation task. He entrusted to us the task of completing the compilation. Professor Tannaka's unfinished manuscript developed class field theory along a path closely connected to algebra theory. Weil's recently published Basic Number Theory adopts a similar method, which has, as he states, 'aged well'. However, the more 'modern' method which utilizes cohomology theory, and was strongly acclaimed by many mathematicians, was now adopted by the new authors, Tamagawa, Satake, Hattori and myself. The four of us then organized a series of 'Seminars on Number Theory' which took place every Sunday at my home. The plan of the volume, the details of each chapter, etc., gradually took definite shape during our discussions. The seminars continued for a long time, during which some of us left Japan; Tamagawa left for America and his place was taken by Fujisaki; Satake also left for America and his place was taken by H. Shimizu. Before our project was completed, Fujisaki also left for America, and I, myself, spent the years 1960-61 abroad. It was, I believe, during this extended absence that Toshia Kusaba assisted us by reading through the manuscripts. The compilation of the last chapter, 'History of Class Field Theory', was assigned to my charge, but, as years passed, my official duties grew heavier. I reached 'retirement age' at Tokyo University in 1967, and the following year was spent teaching in France. Being well aware and regretful of the delay that my absence would incur, before leaving for France I asked Professor Y. Kawada,

PREFACE

vii

Hideo Wada and the editorial staff oflwanami Shoten to continue the compilation of the volume without waiting for the completion of the final chapter. Y. Kawada then entrusted to Mr Wada the final editing of the manuscripts, which were soon printed under the charge of Hideo Arai, an editorial staff member of Iwanami Shoten. The final chapter was completed upon my return to Japan sometime after the main part of the volume had been printed. This volume is, thus, an outcome of the co-operation ofa number of people. We, therefore, wondered under whose name this volume should appear. We consulted Professors Tannaka, Tamagawa, Satake, Hattori, Shimizu and Fujisaki about this matter, and we reached the decision that my name should appear as the editor of the volume. The main part of the volume and Appendix 1 are to be credited to Tamagawa, Satake and Hattori. Chapter I was compiled by Professor Hattori. Tremendous assistance was given by Genjiro Fujisaki who finalized a large part of the manuscript. The section on Weil groups was written by Mr Shimizu. Mr Wada undertook the burden of the final editing of the manuscript and of the proof-reading. Mr Arai took charge of a large part of the matters concerning the editing, printing and publishing. We wish to express our most sincere thanks to all the people who assisted us in the publishing of this volume. The book by Kawada [I), which was published in 1957, elaborates (in Japanese) on class field theory utilizing cohomology theory. The lecture notes by Artin and Tate [V], containing the lectures given during the year 1952-53, were published in 1967. A volume by Weil [VJ, and one by Cassels and Frohlich [I] have also recently been published. Our volume, however, consists of material not completely contained in any one of the abovementioned volumes. However, there remain a number of important subjects such as the Principal Ideal Theorem, complex multiplication, s-functions and other subjects connected to algebraic geometry which are not treated in our volume. It is hoped that a volume which will continue where ours left off will be published. The task of compiling a volume on number theory, more full and complete than the present one, I hope, will be successfully accomplished by the younger generation of mathematicians upon whom the future of number theory rests. Tokyo, Japan

SHOKICHI lYANAGA

Contents Preface .

V

Table of notations

X

CHAPTER I. CoHOMOLOGY OF GROUPS

§1. Tensor products and groups of homomorphisms

§2. §3. §4. §5, §6. §7. §8.

Homology and cohomology . . . Inhomogeneous complexes ~(g) Subgroups and related cohomology groups Tensor products and cup products . Cohomology theory of finite groups Cohomology theory of cyclic groups Tate's theorem and Galois cohomology

.

1 11

25 31 42 49 63

68

CHAPTER II. VALUATION THEORY

§1. Valuations of fields

§2. §3. §4. §5. §6. §7. §8.

Complete fields Archimedean valuations . Non-Archimedean valuations I Non-Archimedean valuations II Hilbert's theory Discriminants and differents (local cases) The differential of formal power series

Ill. ADELE RINGS AND IDELE GROUPS Locally compact groups Locally compact rings Local fields . Adele and idele Extensions of the base field The structure of adele rings The structure of idele groups

80 91

96 102

123 139

162

171

CHAPTER

§1, §2. §3. §4. §5. §6. §7.

178 188 197 208

218 228 244

CHAPTER IV. THE MAIN THEOREMS OF CLASS FIELD THEORY

§1. Cyclotomic fields

§2. §3. §4. §5. §6. §7.

. Kummer fields Power residue symbols and Hilbert norm residue symbols Quadratic number fields and the reciprocity laws for quadratic residues Artin-Schreier fields The theory of infinite Galois extensions Main theorems of class field theory .

254 262 270 282 292 300

307

CHAPTER V. PROOFS OF THE MAIN THEOREMS

§1. Local cases

323

§2. §3. §4. §5. §6.

. Proofs of the conductor theorems The first inequality The second inequality and the existence theorem The Reciprocity Law Weil groups .

339 351

viii

362 378

401

CONTENTS

ix

APPENDIX 1. IDEAL THEORY

§1. Ideals in a Dedekind domain . §2. Discriminants and differents (global cases) §3. Artin-Whaples' theory .

450 470

2. HlsTORY OF CLASS FIELD THEORY §1. From Euclid to Hilbert . . §2. Takagi and Artin's class field theory . §3. The development of the theory after Takagi and Artin

479 494 505

433

APPENDIX

BIBLIOGRAPHY

519

INDEX

537

Table of Notations The following notations are elaborated in Chapter I and are frequently used in Chapter V. A reference such as 4.3 (34) means Chapter I, §4.3, page 34.

a"T = (aT)" A ® 9 B =A® B/S', S' = {a(a ® b) - a® b}gem a(a ® b) = aa ® ab Z(9) ® 9 A ~ A; a® a - aa, Reductive Law

Hom9 (Z(9), A) ~ A; f - f( s), Reductive Law Homli (C, Hom9 (B, A))~ Hom 9 (B ®IJ C, A), Associative Law A 9 = A ® Z(9), 9-regular; (a ® r)a = a ® ra A 9 = Hom(Z(9), A), 9-coregular; (af)r = f(ra) Ali I 9 = Homli(Z(9), A); A is an fJ-module gA = {a EA I Na = O} 9A = {a EA I aa = a, 'vaE9} DA = {aa - a I a E 9, a E A}gen I. We set a,= cb'(l + bryi. Then we have lim,--+ 00 wiciv) ;:::: -wK(IIj) ;:::: -(e -

~

1, v = 1, ... ,f,

1)

(wk is the restriction of wK to k) => CJv E 0.

D

THEOREM 4.8'. Let k be a discrete complete non-Archimedean valuation field, and let K be an extension of k of relative degree n. If the residue class field 5l of K is a separable extension of the residue class field f of k, the valuation ring D of K has as its base over the valuation ring o of k a set consisting of the powers of an element, i.e., there exists an element oc of D such that an arbitrary element 0 of D may be written uniquely as

ai E 0.

Proof We are now given an element oc of D such that the residue class ii generates 5l; i.e., 5l = f(ii.). Let f(X) be a polynomial in o[X] such that

CH.II, §4.4]

NON-ARCHIMEDEAN VALUATIONS I

./(X) E f[X] is an irreducible polynomial with ./('Fi.)

= 0. Furthermore, let

117

wK

denote an exponential valuation of K. Then we have wK(f(ix)) ~ I, and wK(f'(ix)) = 0. We may assume that wK(f(ix)) = I. Indeed, if wK(f(ix)) > I, we may replace ix by {3 = ix + IT; then, since we have f(/3) = f(ix) + Ilf'(ix) + ½Il 2 f"(ix) + •··, wK(Ilf'(ix)) = I, we have wK(f(ix)) = I. We may therefore assume thatf(ix) is a prime element ofK.Hence,byTheorem4.8,theset{ix'.f(ix)1\ 0 ~ i ~f-1, 0 ~j ~ e - l} constitutes an o-base of D, and therefore the set {1, ix, ... , ixn- 1 } constitutes an o-base of D. D 1. The p-adic number field. Let p be a prime number. Then the nonArchi~edean valuation \\ llv introduced in §1.4 is obviously discrete, and the full;Ction wv(a) = -logv \\a \Iv (a E Q*) is the normal exponential valuation belonging to the valuation \\ \Iv, and p is a prime element with respect to t,he valuation. The completion Qv of Q with respect to the valuation \I llv is called the p-adic number field. Denoting the valuation ring and the valuation ideal of Qv by o and p, we have o/µ ~ Z/(p) = the prime field Fv with characteristic p; as a complete set of representatives of the latter quotient set we may choose {O, I, ... , p - I}. Hence, by virtue of Theorem 4.6, a p-adic number a (i.e., an element of Qv) is written uniquely as EXAMPLE

(32)

a=

I

v;;> -

CvPv,

Cv

= 0, 1, ... , p - I.

D

oo

2. The µ-adic number field. Let k be an extension of finite degree of the p-adic number field Qv, and let µ be the prime spot of k which is uniquely determined as the extension of (p) to k; we call this k the µ-adic number field. Let [k: Qvl = n, e = the ramification exponent of µ with respect to Qp, and letf = the relative degree of :p with respect to QP. Then we have ef = n. The normal exponential valuation wP of k is given by EXAMPLE

a Ek*.

(33)

The residue class field f of k is the extension F pf of the prime field F v such that [F pf : F p] = f Therefore the polynomial f(X)

= XPf-1 - 1

118

[CH.II, §4.4

VALUATION THEORY

decomposes into mutually distinct linear factors in f.. Hence, by Hensel's Lemma,f(X) is decomposed into linear factors already in o[X]. The µ-adic number field k therefore contains all the (p 1 - lYh roots of unity which, together with 0, constitute a complete system of representatives of the quotient sett Let { be a primitive (p 1 - l)th root of unity contained ink, and letg(X) be the irreducible monic polynomial in Qp[X] such thatgW = 0. The polynomial g(X) is then a factor of f(X), and therefore its roots are mutually distinct modµ. Hence, by Hensel's Lemma, the reduced polynomial g(X) (=g(X) modp) is also irreducible. Meanwhile, since a root of g(X) = 0 generates f, the degree of g(X), or equivalently of g(X), isf. Let us denote the field QPW by T1 • Then the extension T1/ QP is of degree f, and p is unramified for this extension. Let us now observe the extension k/T1 of relative degree e. By virtue of (21 ), the prime spot of T1 'ramifies completely' for this extension, i.e., its relative degree is 1, and its ramification exponent is e. Let 'TT be a prime element ofµ. Then the ramification exponent of the prime spot of T1 for Thr)/T1 is e, and therefore the relative degree [TtC7r): T1 ] must be a multiple of e. Hence we have k = T 1(7r). Let

x• + a.-1x•- 1 + ... + ao =

0,

be an irreducible equation with a root 'TT. Since we have

wp(a 17r 1)

=i

(mode),

the equality

implies that

wp(a0 ) = e,

wp(a1)

;.,:

e (i

= 1, ... , e -

1),

and therefore we have (34) cx 1 is an integer at the prime spot of T1 (i = 0, .. :, e - 1) and, specifically, cx 0 is a unit. An equation of this form satisfying (34) is called an Eisenstein

equation.

D

3. Fields offormal power series. Let k = F(x) be a rational function field, and let p(x) be an irreducible polynomial in k. Then the valuation II IIP EXAMPLE

CH.II, §4.4]

NON-ARCHIMEDEAN VALUATIONS I

119

of k introduced in §1.4 (II) is discrete. Specifically, let us set p(x) = x, and let k be the completion of k with respect to I !Ix; its valuation ring and the valuation ideal being denoted by o and :p. Then we have o/f, ,. . _, F[x]/(x) ,...._, F, and hence we may choose the field Fas a complete system of representatives of o/f,. By Theorem 4.6 an arbitrary element a(x) of k is uniquely written as (35)

a(x)

=

!

a,x•,

v;!> - oo

Elements a(x) =

!v;1>-oo

a,x•, b(x) =

a(x)

(36)

!v;1>-oo

+ b(x) = !

b,x• of ksatisfy

(a,

+ b,)x•,

v;!>- oo

a(x)b(x)

=

! (!

v;!>- oo

a,,bµ:)x•.

u+u'=v

Conversely, for a given field F, the set of formal power series (35) with addition and multiplication defined by (36) is obviously a field which is called a formal power series field and is denoted by F((x)). The normal exponential valuation Wz of k = F((x)) belonging to I] !Ix is given by

wz(a(x)) = v 0 ,

(37)

4.9. Let k be a discrete complete non-Archimedean valuation field whose characteristic is the same as the characteristic of the residue class field f = o/-p. If, furthermore, every subfield off is perfect, we may choose afield Fas a complete system of representatives off, and we have k = F((x)). The exponential valuation of k is then equivalent to the valuation Wz of F((x)) defined in (37). Proof Since the characteristics of k and f are equal, the prime field contained ink must also be contained in o. Meanwhile, we let F be a maximal subfield contained in o whose existence is now guaranteed by Zorn's Lemma. Since every non-zero elen;ient of F is a unit, the mapping THEOREM

F3 a---+ ii= a (modµ)

E

f

gives an isomorphism from F onto a subfield F of f. It suffices to show that F = f. Suppose that we are given an element x of o with x ¢ F. The element x may not be transcendental over F; indeed, if it were so, we would have F[.x] n v = {O}, and hence every non-zero element of F[x] is a unit, which implies thatF ;= F (x) ~ o, contradicting our assumption that Fis

x thus satisfies an irreducible equation in F, X + c1 xn-l + "' + C = 0, C1 EF.

maximal in o. The element /(X) =

11

11

120

[CH.II, 4.5

VALUATION THEORY

By our assumption on the perfectness of subfields off, the equation/(X) = 0 has no multiple roots. Hence the polynomial f(X) decomposes in f[X] as /(X)

= (X - x)li(X),

and the factors ( X - x) and Ii( X) are relatively prime. Hence, by Hensel's Lemma, f(X) = 0 has a root x1 in o. If n > 1 we have F ~ F(x 1) ~ o contradicting our assumption of the maximality of F. Hence F must be equal to f.

D In Theorem 4.9 the assumption on the perfectness of subfields off always holds if the characteristic off is Oor iff is a finite field. Subsequently, we shall see that the assumption may be weakened to that of the perfectness of f alone (Corollary of Theorem 4.10). 4.5. The system of multiplicative representatives

Let k be a discrete complete non-Archimedean valuation field. If k has nonzero characteristic p, its residue class field f is also of characteristic p. When the characteristic of k is 0, k contains the rational number field Q. If in this case an exponential valuation of k, when restricted on Q, is equivalent with the p-adic valuation wp, then we have pl == 0 (mod :p), and hence f has characteristic p. Whereas, if the exponential valuation w, when restricted on Q, is trivial, then we have w(nl) = 0, nl '¢. 0 (mod :p ), and hence the characteristic off is 0. 4.10. Let k be a discrete complete non-Archimedean valuation field. Suppose that the residue class field f of k is a perfect field of characteristic p =f 0. Then there exists a set W having the following properties: (i) WC 0, (ii) F = Wu {O} is a complete system ofrepresentatives of i (hence W cf, 0), (iii) a,f] E W ~ af],a- 1 E W, (iv) WP= W. The set W is uniquely determined by (i), (ii) and (iv). If, furthermore, the characteristic of k is also p, then we have: (v) F = wu {O} is a.field which is isomorphic to f. Proof Let us choose an element a0 from each residue class C of f. Since the field f = o/:p is perfect, there exist elements a 1 , a2 , ••• , an, ... of o such that THEOREM

(38)

af

hold. Setting bn

=

=a

0,

a~

=a

1,

af, we have a0

... ,

=

a~

=an-

1 , ...

(mod :p)

b 0 , and all bn belong to the class C.

CH.II, §4.5]

NON-ARCHIMEDEAN VALUATIONS I

121

We shall show that the sequence {bn} is a Cauchy sequence. For this purpose, we shall first show the following general property: (39)

a,b E o, a

= b (mod pn) for n

;,, I

=?-

av

= bP (mod pn+ 1).

Indeed, by the hypothesis we have a= b

+ C7Tn,

c E o,

7T

is a prime element;

hence we have av = (b = bP

+ C7Tn)v + pb K. Let cp be a valuation

CH.II, §5.5]

NON-ARCHIMEDEAN VALUATIONS II

137

of k, let q/j1, .. ., 'Pt) be of its extensions to K, and let