Local Class Field Theory - University Of British Columbia .
Local ClassField TheoryKENKICHI IWASAWAPrinceton UniversityOXFORD UNIVERSITY PRESS New YorkCLARENDON PRESS - Oxford1986
Oxford University PressOxford New York TorontoDelhi Bombay Calcutta Madras KarachiPetaling Jaya Singapore Hong Kong TokyoNairobi Dar es Salaam Cape TownMelbourne Aucklandand associated companies inBeirut Berlin Ibadan NicosiaCopyright C) 1986 by Kenkichi IwasawaPublished by Oxford University Press, Inc.,200 Madison Avenue, New York, New York 10016Oxford is a registered trademark of Oxford University PressAll rights reserved. No part of this publication may be reproduced,stored in a retrieval system, or transmitted, in any form or by any means,electronic, mechanical, photocopying, recording, or otherwise,without the prior permission of Oxford University Press.Library of Congress Cataloging-in-Publication DataIwasawa, Kenkichi, 1917–Local class field theory.(Oxford mathematical monographs)Bibliography: p.Includes index.1. Class field theory.I. Title.II. Series.QA247.195413 1986512'.7485-28462ISBN 0-19-504030-9British Library Cataloguing in Publication DataIwasawa, KenkichiLocal class field theory.—(Oxford mathematical monographs)1. Fields, Algebraic I. Title512'.3 QA247ISBN 0-19-504030-9246897531Printed in the United States of Americaon acid-free paper
PrefaceLocal class field theory is a theory of abelian extensions of so-calledlocal fields, typical examples of which are the p-adic number fields. Thisbook is an introduction to that theory.Historically, local class field theory branched off from global, or classical,class field theory, which studies abelian extensions of global fields—that is,algebraic number fields and algebraic function fields with finite fields ofconstants. So, in earlier days, some of the main results of local class fieldtheory were derived from those of the global theory. Soon after, however,in the 1930s, F. K. Schmidt and Chevalley discovered that local class fieldtheory can be constructed independently of the global theory; in fact, theformer provides us essential devices for the proofs in the latter. Around1950, Hochschild and Nakayama brought much generality and clarity intolocal class field theory by introducing the cohomology theory of groups.Classical books such as Artin [1] and Serre [21] follow this cohomologicalmethod. Later, different approaches were proposed by others—for example, the method of Hazewinkel [11], which forgoes cohomology groups,and that of Kato [14], based on algebraic K-theory. More recently,Neukirch also introduced a new idea to local class field theory, whichapplies as well to global fields.Meanwhile, motivated by the analogy with the theory of complexmultiplication on elliptic curves, Lubin and Tate showed in their paper [19]of 1965 how formal groups over local fields can be applied to deduceimportant results in local class field theory. In recent years, this idea hasbeen further pursued by several mathematicians, in particular by Coleman.Following this trend, we shall try in this book to build up local class fieldtheory entirely by means of the theory of formal groups. This approach,though not the shortest, seems particularly well suited to prove some deepertheorems on local fields.In Chapters I and II, we discuss in the standard manner some basicdefinitions and properties of local fields. In Chapter III, we consider certaininfinite extensions of local fields and study formal power series withcoefficients in the valuation rings of those fields. These results are used inChapters IV and V, where we introduce a generalization of Lubin–Tateformal groups and construct similarly as in [19] abelian extensions of localfields by means of division points of such formal groups. In Chapter VI, themain theorems of local classfield theory are proved: we first show that theabelian extensions constructed in Chapter V in fact give us all abelianextensions of local fields, and then define the so-called norm residue mapsand prove important functorial properties of such maps. In Chapter VII, theclassical results on finite abelian extensions of local fields are deduced fromthe main theorems of Chapter VI. In the last chapter, an explicit reciprocitytheorem of Wiles [25] is proved, which generalizes a beautiful formula ofArtin–Hasse [2] on norm residue symbols.
viPrefaceThe book is almost self-contained and the author tried to make theexposition as readable as possible, requiring only some basic background inalgebra and topological groups on the part of the reader. The contents ofthis book are essentially the same as the lectures given by the author atPrinceton University in the Spring term of 1983. However, the originalexposition in the lectures has been much improved at places, thanks to theidea of de Shalit [6].In 1980, the author published a book [13] on local class field theory inJapanese from Iwanami-Shoten, Tokyo, which mainly followed the idea ofHazewinkel fil]. When the matter of translating this text into Englisharose, the author decided to rewrite the whole book in the manner justdescribed. In order to give the reader some idea of other approaches inlocal class field theory, a brief account of cohomological method andHazewinkel's method are included in an Appendix. At the end of the book,a short list of references is attached, containing only those items in theliterature mentioned in the text; for a more complete bibliography on localfields and local class field theory, the reader is referred to Serre [21]. Anindex and a table of notations are also appended for the convenience of thereader.The author expresses here his hearty gratitude to D. Dummit and E.Friedman who carefully read the book in manuscript and offered manyvaluable suggestions for its improvement. He also thanks D. W. Degenhardt of Oxford University Press for his help in publishing this book.PrincetonSeptember 1985K.I.
ContentsChapter I. Valuations31.1. Some Basic Definitions371.2. Complete Fields1.3. Finite Extensions of Complete FieldsChapter II. Local Fields2.1.2.2.2.3.2.4.2.5.121818General Properties22The Multiplicative Group k x25Finite ExtensionsThe Different and the Discriminant32Finite Galois Extensions29Chapter III. Infinite Extensions of Local Fields353.1. Algebraic Extensions and Their Completions353.2. Unramified Extensions and Totally Ramified Extensions3.3. The Norm Groups403.4. Formal Power Series433.5. Power Series over ok45Chapter IV. Formal Groups FAX, Y)4.1.4.2.4.3.4.4.Formal Groups in GeneralFormal Groups Ff(X, Y)The o-Modules W'fl57Extensions Lnig61505053Chapter V. Abelian Extensions Defined by Formal Groups5.1. Abelian Extensions Ln and lc"655.2. The Norm Operator of Coleman695.3. Abelian Extensions L and lc,75Chapter VI. Fundamental Theorems6.1. The Homomorphism Pk846.2. Proof of L k lcab6.3. The Norm Residue Map3680888065
viiiContentsChapter VII. Finite Abelian Extensions7.1.7.2.7.3.7.4.Norm Groups of Finite Abelian Extensions98Ramification Groups in the Upper Numbering101The Special Case k7'nlk107110Some ApplicationsChapter VIII. Explicit Formulas8.1.8.2.8.3.8.4.8.5.a-Sequences116The Pairing (a, 13 )The Pairing [a, O ],The Main TheoremThe Special Case for k120123127 Qp116133137AppendixA.1. Galois Cohomology Groups137A.2. The Brauer Group of a Local FieldA.3. The Method of Hazewinkel146Bibliography151Table of NotationsIndex98155153141
LOCAL CLASS FIELD THEORY
Chapter IValuationsIn this chapter, we shall briefly discuss some basic facts on valuations offields which will be used throughout the subsequent chapters. We shallfollow the classical approach in the theory of valuations, but omit the proofsof some elementary results in Section 1.1, which can be found in manystandard textbooks on algebra.t For further results on valuations, we referthe reader to Artin [1] and Serre [21].1.1.Some Basic DefinitionsLet k be a field. A function v(x) on k, x E k, is called a valuation of k if itsatisfies the following conditions:(i) v(x) is a real number for x *0; and v(0) GC.(ii) For any x, y in k,min(v(x), v(y)) Is v(x y).(iii) Similarly,v(x) v(y) v(xy).EXAMPLE 1.Letv0(0) 00, vo(x) 0 for all x *0 in k.Then vo is a valuation of k. It is called the trivial valuation of k.EXAMPLE 2. Let p be a prime number. Each non-zero rational number xcan be uniquely written in the form x pey, where e is an integer and y is arational number whose numerator and denominator are not divisible by p.We define a function vp on the rational field Q byv(0) 00;vp (x) e, if x *0 and x pey as above.Then vp is a valuation of Q; it is the well-known p-adic valuation of therational field. vp is the unique valuation v on Q satisfying v(p) 1.ttEXAMPLE 3. Let F be a field and let F(T) denote the field of all rationalfunctions in an indeterminate T with coefficients in F. Then each x *0 inF(T) can be uniquely written in the form x Tey, where e is an integer andy is a quotient of polynomials in F[T], not divisible by T. Putting v(0) 00and v(x) e for x *0 as above, we obtain a valuation v of the fieldk F(T) such that v(T) 1. Instead of T, we can choose any irreduciblet See, for example, Lang [16] and van der Waerden [23].tt Compare van der Waerden [23].
4Local Class Field Theorypolynomial f(T) in F(T) and define similarly a valuation v on k F(T)with v(f) 1.Let v be a valuation of a field k. Then it follows immediately from thedefinition thatv(x) v(y) v(x y) v(x).v( 1) 0, v(–x) v(x),Here 1 ( 1 k ) denotes the identity element of the field k. Leto {x I x E k, v(x) 0},p I X E k, v(x) O}.Then o is a subring of k continuing 1 and p is a maximal ideal of o so thatis a field. o, p, and f are called the valuation ring, the maximal ideal, andthe residue field of v, respectively. By (iii) above, the valuation v defines ahomomorphismv:k x –*R from the multiplicative group le of the field k into the additive group R ofreal numbers. Hence the image v(k x ) is a subgroup of R , and we havele 1 U v(kx)whereU Ker(v) {xE, v(x) 0}.U is called the unit group of the valuation v.Let v be a valuation of k. For any real number a 0, define a function,u(x) on k by,u(x) crv(x),for all xEk.Then /.2 is again a valuation of k. When two valuations v and p, on k arerelated in this way—namely, when one is a positive real number times theother—we writevand say that v and p. are equivalent valuations of k. Equivalent valuationshave the same valuation ring, the same maximal ideal, the same residuefield, the same unit group, and they share many other important properties.Let v be a valuation of a field k. For each x E k and a E R, letN(x, a) {y I yEk, v(y – x) a } .This is a subset of k containing x. Taking the family of subsets N(x, a) forall a E R as a base of neighbourhoods of x in k, we can define a Hausdorfftopology on k, which we call the v-topology of k. k is then a topologicalfield in that topology; the valuation ring o is closed in k and the maximalideal p is open in k. A sequence of points, x 1 , x2 , x3 , . . . , in k converges to
5Valuationsx E k in the v-topology—that is,lim xn x,n- 00if and only iflim v(xn — x) X.When this is so, thenbin v(x) V(X).In fact, if x * 0, then v(x) v(x) for all sufficiently large n. A sequencex l , x2 , x 3 , . . . in k is called a Cauchy sequence in the v-topology whenv(x,n — xn)—* 00,as m, n--- 00 A convergent sequence is of course a Cauchy sequence, but the converse isnot necessarily true. The valuation v is called complete if every Cauchysequence in the v-topology converges to a point in k. If v is complete thenthe infinite sum00E xn hill EXnn 1converges in k if and only ifv(xn )-- 00,as n -- oo.A valuation v of k is called discrete if v(k x ) is a discrete subgroup ofIt—that is, ifv(k x ) Zi3 Inf3 I n 0, 1, 2,. . .1for some real number f3 a 0. If p 0, then v is the trivial valuation vc, inExample 1. When )3 1—that is, whenv(V) Z {0, 1, 2, . .},v is called a normalized, or normal, valuation of k. It is clear that avaluation v of k is discrete but non-trivial if and only. if v is equivalent to anormalized valuation of k.Let k' be an extension field of k, and 3/' a valuation of k'. Let y' 1 kdenote the function on k, obtained from y' by restricting its domain to thesubfield k. Then y' 1 k is a valuation of k, and we call it the restriction of y'to the subfield k. On the other hand, if v is a valuation of k, any valuationy' on k' such thatI, ' 1 k yis called an extension of v to k'. When y' on k' is given, its restriction I, ' 1 kis always a well-determined valuation of k. However, given a valuation v onk, it is not known a priori whether I/ can be extended to a valuation y' of k'.The study of such extensions is one of the main topics in the theory ofvaluations.
Local Class Field Theory6Let I, ' I k v as stated above. Then the v'-topology on k' induces thev-topology on the subfield k so that k is a topological subfield of k'. Let o',and f' denote the valuation ring, the maximal ideal, and the quotientfield of v', respectively: {.7C' Ep' k' I v'(x')a-0},k' I v' (x') O}, 07p ,.Thenp p'nk p'noo o'nk,so thatf o/p o/(p'n 0) (0 p')Ip' ç o'ip' f'.Thus the residue field f of v is naturally imbedded in the residue field f' ofv'. On the other hand, if' I k v also impliesv(k x ) v'(k'')R .Lete e(v' I v) [v'(le x ): v(0)],f f(v' I v) [f' :where [vi(le x ):(k x )] is the group index and [f' : f] is the degree of theextension f1f. e and f are called the ramification index and the residuedegree of v'/v, respectively. They are either natural numbers 1, 2, 3, . . . , or 00.The following proposition is a fundamental result on the extension ofvaluations.PRoposmoN 1.1. Let v be a complete valuation of k and let k' be analgebraic extension of k. Then v can be uniquely extended to a valuation y'of k': 3/' I k v. If, in particular, k' I k is a finite extension, then 1/' is alsocomplete, and1v'(x') – v(N vik (x)),nfor all x'Ek',where n [k' :k] is the degree and N kvk is the norm of the extension k' I k. Proof. We refer the reader to van der Waerden [23].Let k' I k, v, and y' be as stated above and let a be anautomorphism of k' over k. Then v'o a v'—that is,COROLLARY.vi (o-(x')) v'(x'),for all x'v(o') o',cr(p') p'.Ek',so thatHence a is a topological automorphism of k' in the v'-topology, and itinduces an automorphism a' of f' over f:J' . 1 2- Proof.The proof can easily be reduced to the special case where k' I k is
7Valuationsa finite extension. The first part then follows from N kyk (a(x 1 )) N kik(x 1 ) The second part is obvious.Let v be a valuation of k, not necessarily complete. It is well known thatthere exists an extension field k' of k and an extension V of v on k' suchthat V is complete and k is dense in k' in the v'-topology of k'. Such a fieldk' is called a completion of k with respect to the valuation v. Moreprecisely, we also say that the pair (k', v') is a completion of the pair (k, v).Let (k", v") be another completion of (k, v). Then there exists a kisomorphism u: k' 2, k" such that V v" 0 a. Thus a completion is essentiallyunique, and hence (k', v') is often called the completion of (k, v). By thedefinition, each x' in k' is the limit of a sequence of points, x i , x2 , . . . , in kin the 'V-topology:-x' lim xn .n--.00ThenV f (X 1 ) liM V f (X n ) liM V (X n)*rt--.00Hence if x' *0, then v'(x ') v(x) for all sufficiently large n. It follows that3/ 1 (lex) v(e),fi f,so thate(v' I v) f(v' I v) 1in this case. It is also clear that if (k', v') is a completion of (k, v) and ifti cry, a 0, then (k', te), with if cry', is a completion of (k, it).4. Let K be an extension of k and let /2 be an extension of v on kto the extension field K:ttIk v. Suppose that tt is complete. Let k'denote the closure of k in K in the s-topology. Then k' is a subfield of K,and (k', v'), with V ti I k', is a completion of (k, v).EXAMPLETo study a valuation v on a field k, we often imbed (k, v) in itscompletion (k', v'), investigate the complete valuation V, and then deducefrom it the desired properties of v. For example, in this manner we candeduce from Proposition 1.1 that if K is an algebraic extension of k, thenevery valuation on k has at least one extension on K.1.2.Complete FieldsLet v be a valuation of a field k. We say that k is a complete field withrespect to v, or, simply, that (k, v) is a complete field, if v is a complete,normalized valuation of k. tLet v be a normalized valuation of a field k, not necessarily complete,and let (k', v') be the completion of (k, v). Then v 1 (k") v(kx) Z byt Some authors call a field a complete field if it is associated with a complete valuation, notnecessarily normalized.
8Local Class Field TheorySection 1.1. Since 3/' is complete, (k', V) is a complete field. Many naturalexamples of complete fields are obtained in this manner.EXAMPLE 5. Let p be a prime number and let vp be the p-adic valuation ofthe rational field Q in Example 2, Section 1.1. Since vp is a normalizedvaluation, the completion (k', v') of (Q, vp ) is a complete field. k' isnothing but the classical p-adic number field Qp , and vi, often denotedagain by vp , is the standard p-adic valuation of Qp . For (Qp , vp ) , thevaluation ring is the ring Zp of p-adic integers and the maximal ideal is pZpthe residue field is Zp /pZp Fp , the prime field with p elements. sothaNote that V(p) 1.EXAMPLE 6. Let F be a field, T an indeterminate, and F((T)) the set of allformal Laurent series of the formE— 00an Tn,an E F, nwhere — 00 n indicates that there are only a finite number of terms an Tnwith n O, an O. Then k F((T)) is an extension field of F in the usualaddition and multiplication of Laurent series. Let v(0) 00 and letv(x) i if00xo,x E an Tn,with ai O.n iThen one checks easily that (k, v) is a complete field. The valuation ring isthe ring F[[T]] of all (integral) power series in T over F, the maximal idealis TF[[x]], the residue field is F[[T]]ITF[[T]]- F, and v(T) 1. The field kcontains the subfield F(T) of all rational functions of T with coefficients inF, and the restriction v F(T) is the normalized valuation of F(T) inExample 3, Section 1.1. Furthermore, (k, v) is the completion of(F(T), yI F(T)).Now, let (k, v) be any complete field and let o, p, and oip denote,respectively, the valuation ring, the maximal ideal, and the residue field ofthe valuation V. They are also called the valuation ring, and so on, of thecomplete field (k, v). Since v(k x ) Z, there exists an element r in k suchthatv(z) 1.Any such element r is called a prime element of (k, v). Fix jr. Sincep {x E k v(x) % 1} in this case, (z) oz.Hence, for any integer n 0,p n (.70) 0.70 {xEk I v(x)n}.In general, let a be an o-submodule of k, different from {0}, k. As a* k,the set {v(x) I x E a, x *0 } is bounded below in Z, and if n denotes the
9Valuationsminimum of the integers in this set, thena {X Ek I v(x)n} Og n .Such an o-submodule a of k, a* {0}, k, is called an ideal of (k, v). The setof all ideals of (k, v) forms an abelian group with respect to the usualmultiplication of o-submodules of k. By the above, it is an infinite cyclicgroup generated by p.An ideal of (k, v), contained in o, is nothing but a non-zero ideal of thering o in the usual sense. Hence the sequence{0}pnp2p p0 0(1.1)gives us all ideals of the ring o. Since prz (Tun), o is a principal idealdomain. Furthermore, in this case v(x) n #.v(x) n —1 for any n E Z.Hence all the ideals If, 0, in (1.1) are at the same time open and closedin k, and they form a base of open neighbourhoods of 0 in the v-topology ofk. Since pn (Tun)* {0}, the field k is totally disconnected and non-discreteas a topological space.Next, we consider the multiplicative group k x of k. Let (g) denote thecyclic subgroup of k x , generated by a prime element Tr. Then theisomorphism kx I U v(k x ) Z in Section 1.1 impliesk (it) x U,(n)Z.LetUo U,Un 1 pn {x Eolx 1 mod Pnfor
Local class field theory is a theory of abelian extensions of so-called local fields, typical examples of which are the p-adic number fields. This book is an introduction to that theory. Historically, local class
This is a writeup of my Master programme course on Quantum Field Theory I (Chapters 1-6) and Quantum Field Theory II. The primary source for this course has been ‹ Peskin, Schröder: An introduction to Quantum Field Theory, ABP 1995, ‹ Itzykson, Zuber: Quantum Field Theory, Dover 1980, ‹ Kugo: Eichtheorie, Springer 1997,
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