CLASS FIELD THEORY EMILARTIN
JOHN TATE
AMS CHELSEA PUBLISHING American Mathematical Society • Providence:, Rhode Islan...
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CLASS FIELD THEORY EMILARTIN
JOHN TATE
AMS CHELSEA PUBLISHING American Mathematical Society • Providence:, Rhode Island
2000 Mathematics Sullject Clas8i/ic4tion. Primary llR37; Secondary 1101, llR34.
For additional information and updates 011 this book, visit www.ams.org/bookpages/cbel366
Library of Congresa CataloginginPublicatlon Data Artin, Emil, 18981962. Class field theory / Emil Artin, John Tate. Originally published: New York: W. A. Benjamin, 1967. Includes bibliographical references. ISBN 978()"82184426·7 (a1k. paper) 1. Class field theory. I. Tate, JaM 'Ibm!nc:e, 1925 joint author.
n. Title.
QA247.A75 2008 512.7'4dc22
2008042201
Copying and reprinting. IndividueJ teadert of this publicatioa, &Ild nonprofit libraries acting fur them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication Is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Chari"" Street, Providence, Rhode Island 029042294, USA. Requests can also be made by email to reprilltpermissionillams. org.
© 1967, 1990 held by the American Mathematical Society.
AU rights reserved. Reprinted with oorrecttons by the American Mathematkal Society, 2009. The American Mathematical Society retains ..u rights except thooe granted to the L'nited States Government. Printed in the United States of America..
@ The paper UBed in this book is acid·free and falls within the guidelinell established to ensure permanence and durability. Visit the AMS home page at http://vwv.aDls.Qrg/ 10 9 8 76 5 4 3 2 1
14 13 12 11 10 09
Contents Preface to the New Edition Preface
Preliminaries 1. Idcles and Idcle Classes 2. Cohomology 3. The Herbraud Quotient 4. Local Cla.ss Field Theory Chapter V. The First Fundamental Inequality 1. Statement of the First Inequality 2. First Inequality in FUnction Fields 3. First Inequality in Global Fields 4. Consequences of the First Inequality Chapter VI. Second Fundamental Inequality 1. Statement and Consequences of the Inequality 2. Kummer Theory 3. Proof in Kummer Fields of Prime Degree 4. Proof in pextensions 5. Infinite Divisibility of the Universal Norms 6. Sketch of the Analytic Proof of the Second Inequality
Chapter VII. Reciprocity Law 1.
2. 3. 4.
Introduction Reciprocity Law over the Rationals Reciprocity Law Higher Cohomology Groups in Global Fields
v
vii
1 1 3 5 8 11
11 11
13 16 19 19
21 24 27
32 33
35 35 36 41
52
55
Chapter VIII. The Existence Theorem 1. Existence and Ramification Theorem 2. Number Fields 3. FUnction Fields 4. Decomposition Laws and Arithmetic ProgressioD8
62
Cha.pter IX. Connected Component of Idcle Classes 1. Structure of the Connected Component 2. Cohomology of the Connected Component
65 65 70
Chapter X. The Grunwa.ldWang Theorem 1. Interconnection between Local and Global mth Powers
73
55
56 59
73
CONTENTS
2.
Abelian Fields with Given Local Behavior
'.
3. Cyclic Extensions Chapter XI. Higher Ramification Theory 1. Higher Ramification Groups 2. Ramification Groups of a Subfield 3. The General Residue Class Field 4. General Local Class Field Theory 5. The Conductor Appendix: Induced Characters
83 83 86 90
92 t.
Chapter XII. Explicit Reciprocity Laws 1.
2. 3. 4.
76 81
Formalism of the Power Residue Symbol Local Analysis Computation of the Nonn Residue Symbol in Certain Local K ummel' Fields The Power Reciprocity Law
99
104 109' 109
III 114 122
Chapter XIII. Group Extensions 1. Homomorphisms of Group Extensions 2. Commutators and Transfer in Group Extensions 3. The Akizuki Witt Map v: H2{G,A) > H2(G/H,AH) 4. Splitting Modules and the Principal Ideal Theorem
127 127 131
Chapter XIV. Abstract Class Field Theory 1. Formations 2. Field Formations. The Brauer Groups 3. Class Formations: Method of Establishing Axioms 4. The Main Theorem Exercise 5. The Reciprocity Law Isomorphism 6. The Abstract Existence Theorem
143 143 146 150 154 i 57
Chapter XV.
167
Bibliography
Weil Groups
134 137
158 163
191
Preface to the New Edition The origInal preface which follows tells about the history of these notes and the missing chapters. This book is a slightly revised edition. Some footnotes and historical comments have been added in an attempt to compensa.te for the lack of references and attribution of credit in the original. There are two mathematical additions. One is a sketch of the analytic proof of the second inequality in Chapter VI. The other is several additional pages on Weil groups at the end of Chapter XV. They explain that what is there called a Weil group for a finite Galois extem;ion K / F lacks an essential feature of a Weil group in Weil's sense, namely the homomorphism WK,F ..... Gal(K&b/F), but that we recover this once we construct a Weil group for F / F by passing to an inverse limit. There is aloo a sketch of an abstract vel'5ion of Weil's proof of the existence and uniqueness of his WK.F for number fields. I have not renumbered the chapters. After some prelimlnaries, the book still starts with Chapter V, but the mysterious references to the missing chapters have been eliminated. The book is now in TeX. The handwritten German letters are gone, and many typographical errors have been corrected. I thank Mike Rosen for his help with that effort. For the typos we've missed and other mistakes in the text, the AMS maintains a Web page with a list of errata at
http://www.am3.org/bookpages/chel366/ I would like to thank the AMS for republishing this book, and especially Sergei Gelfand for his patience and help with the preparation of the manuscript. For those unacquainted with the book, it is a quite complete account of the algebraic (as opposed to analytic) aspects of classical class field theory. The first four chapters, VVIII, cover the basics of global class field theory, the cohomology of ideIe cla8ses, the reciprocity law and existence theorem, for both number fields and fWlction fields. Chapters IX and X cover two more special topics, the structure and cohomology of the connected component of 1 in the ideIe class group of a number field, and questions of local vs. global behavior surrounding the GrunwaldWang theorem. Then there are two chapters on higher ramification theory, generalized local classfield theory, and explicit reciprocity laws. This material is beautifully covered aloo in [21]. For a recent report, see [8]. There is a nice generalization of our classical explicit formula in [13]. The last three chapters of the book cover ab!tract class field theory. The cohomological algebra behind the reciprocity law is common to both the local and global class field theory of number fields and fWlCtioll fields. Abstracting it led to the definition of a new algebraic structure, 'class formation', which embodies the common features of the four theories. The difference is in the proofs that the id.el.e classes globally, and the multiplicative groups locally, satisfy the axioms of a cla8s formation. Chapter XIV concludes with a discussion of the reciprocity law and existence thoorem for an abstract class formation. In the last
.
chapter XV, Well groups are defined Cor finite 1f1Y'ln. of an IU'hiLrlU")' dAA'! forml\t,iolf, and then, for topological class formatioD~ sat.isfyillR (:tlrtn.iIl·'LXiuIUH wiai('h hold ill the classical cases, a Weil group for the whole formation is c(.IIISlrul:tt'fi, by passage to an inverse limit, The class formation can b~ mc:uvl!rcocl frolll illl W.·jJ group, and the topological groups which occur as Wei! grou~ a.re dlllrlld,f:rl~('c1 hy 1LX11l1llH. The mathematics in this book is the result uf IL l."elltury of dl!vl'l()PCIIICDt, roughly 18501950. Some history is discussed by 11,_, in [51 IUIII in I'K'!vl'ral of the papers in [18). The high point carne in the 192()'H with TIlbf,I'M pr(Mlf t.hlit the finite abelian extensions of a number field are in nalllrHl Ol\(~ to· OIl!' (;urrc:l!polidelll;e with the quotients of the generalized ideal c18.'Ili grolll'" or that fidd, lUll! Artin's proof several years later that an abelian GaJoi::i group /\IId the COl rl'llpnnding irloaJ class group are canonically isomorphic, by an isomorphislII whkh ilTllllu'C\lill known reciprocity laws. The flavor of this book is strollRly illlhwlJ(:1!f1 hy till: !llIll tltcp~ in that history. Around 1950, the systematic usc of th(' l'ohornolOR.Y of groups by Hochschild, Nakayama and the authors shed Dew light. It. nli/Lilll... 1 many thl'Orerns of the local class field theory of the 1930's to be transfprFCoc\ to thro gklllll.l tht'Ory, Ilnd led to the notion of class formation embodying the I:OlIImUIi fml.t.IIWH of IMlt.h theories. At about the same time, Well conceived tht: irleu of Wt'il II:ruullII "nd proved their existence. With those two developments it is fwr to Hay that the c1essicaJ on~dimensional abelian class field theory had rf'.ar.hNl flill IIIlItllrity. ThtlTf' were still a few things to be worked out, such as the loralaud glub"" duulity theuries, and the cohomology of algebraic tori, but it was time for new dirr.(:liolili. They soon came. For example: • • • • • •
Higher dimensional class field theory; Nonabelian reciprocity laws and' the LUKlandll program; Iwasawa theory; Leopold's conjecture; Abelian (and nonabelian) tadic repl'eIIeDtatiOnBj LubinTate local theory, Hayes explicU theory for fuuctJoa Beida, Drinfeld modules; • Stark conjecturesj • Serre conjectures (now theorems). Rather than say more or give references for these, I lIillll)ly r()(~nuullnd what has become a universal reference, the internet. Searchilll/; £Lily of the above topics is rewarding. John Th.U! Scpt.embt:r 2008
.
.' . '
Preface This is a chunk of the notes of the ArtinTate seminar on class field theory given at Princeton University in 19511952, namely the part dealing with global clWiis field. theory (Chapters V through XII) and the part dealing with the abstract theory of class formations and Weil groups (Chapters XIIIXV). The fmlt four chapters, which are not included, covered the cohomology theory of groups, the fundamentals of algebraic number theory, a preliminary discussion of class formations, and local class field theory. In view of these missing sections, the reader will encounter missing references and other minor flaws of an editorial nature, and also some unexplained notations. We have written a few pages below recalling some of these notations and outlining the local class field theory, in an attempt to reduce the "prerequisites" for reading these notes to a basic knowledge of the cohomology of groups and of algebraic theory, together with patience. The reaoon for the long delay in publication was the ambition to publish a revised and improved version of the notes. This new version was to incorporate the advances in the cohomology theory of finite groups which grew out of the seminar and which led to the determination of the higher cohomology groups and to a complete picture of the cohomological aspects of the situation, as outlined in Tate's talk at the Amsterdam Congress in 1954. However this project was never completed and thus served only to prevent the publication of the IIIost important part of the semilUU', namely Chapters V through XII of tbese notes. That this material finally appears is due to the energies of Serge Lang, who took the original notes, continued to urge their publication, and has now made the arrangements for printing. It is a pleasure to express here our appreciation to him for these efforts.
Two excellent general treatments of class field theory, which complement these notes, have appeared during the past year, namely: Cassels and FrOhlich, Algebraic Number Theory, Academic Press, London, 1967. (Distributed in the U.S. by the Thompson Publishing Company, Washington, D.C.).
Weil, Basic Number Theory, SpringerVerlag, Berlin/Heidelberg/New York, 1967.
vi
·,
Preliminaries 1. Ideles and Idele ct. . .
A global field is either a number field of finite dl1gToo owr th .. rational field a function field in one variable over a finite (:mull,lln!. fi(·ld. SIII:h fil!lds have primes p, and corresponding canonical ah:.;olut.e valUllll I I,. fUI whil'll LIll! product formula holds. A local field is the completion k~ of 11 globul lil'lt.! k at n prime p. Thus a local field is either the real field JR, the ES
IPlp
IPts IPlp
and consequently
Hr(a, h,s) R: IT w( a, IT K~) Now the operation of
x
'PIp
PES
II w( a, IT U'P)' p¢S
IPlp
a on DlPlp K~
permutes the factors, and the subgroup of consisting of the elements which carry a given factor K~, into itself is the decomposition group Gcp of!p. It follows that TI'PI" K~ is the Gmodule "induced" by the Gcpmodule K~ and the cohomological theory of induced modules (some time referred to as Shapiro's Lemma, referred to in these notes by the catchword ,emilocal theory) shows that we have isomorphisms
a
W(a, IT Kir) R: Hr(G'J. Kir) CPI, for any fixed prime
I.P dividing p, and similarly HI' (G,
11 UIP)
R.:
W(GIP, U'P)'
'Jlp
These lsomorphisms are canonical, coming from the restriction from G to G'J and the projection of the !pfactor. By the theory of local fields, we have Hr(G'P, U'P) ofor r > 0 if K Al ::> A2 :J ... he a decrea&ing sequence of subgroups invariant under G, and which shrink to the identity in the sense that for each neighborhood U of 1 in A, there is an index i lUck that Ai C U. If HT(G, At/AHd = 0 for all i and some r, then Hr(G, A) = O. LEMMA.
continuotlSly
011
(In characteristic 0, one could avoid the preceding construction by taking a sufficiently small neighborhood of 0 in the additive group of K, and mapping it onto a neighborhood of 1 in K* by means of the exponential fWlction.) For cyclic layers KIF of prime degree p different from the characteristic, one can also deduce hZ{l(GKjF, K*) = P from Theorem q.4 above, using the equations p . p /}(J,p(K·) = Ipl" and qo,p(F*) = lPl"
4. LOCAL CLASS FIELD THlOkY
•
obtained in our computation of the power index (XO : KO,) aWvt!. Finally, one could ignore the second inequality ('Olllplpl,·ry hy provin,; directly that every 2dimensional class has an Wlramified splitt.ing fif'hl, or ""hAt is the same, that the Brauer group of the maximal unramififlcl I'xtmll,ifln of k III trivial (cf. for example [15)). To complete the proof that our formation of mult.iplirl\livr gmuJlII of local fields is a class formation, we must establish Axiom II' of ChApt.pr XIV. For this, we consider the unramified extension KIF of dej!;r«:c~ n. Sinn' tllC' FI'lIiduc dass field is finite, the Galois group G K/ F is cyclic, with a ca..nollica.l K,'ul!ralor, lhe Ftobenius automorphism q; = 'PKIF' For any normal layer K / F, unramified or not, the ex",,:l lI«'Iquence 0+ UK
+
KO
+
Z + 0
yields, on passage to cohomology, po = K oG + Z
+
HI(GK/P,UK) ..... H1(GK/I',K·)  0,
from which we see that HI (G K/ F, UK) is isomorp hie 1.0 the c:olwrncl of FO ..... Z, i.e. is cyclic of order equal to the ramification index r.K/F, b('('aullt' Z hl!re repreoents the value group of K. Thus, for our unramified K/ F, M: haY\' II I (G K IF, UK) = O. On the other hand, we have HO(GK1F,l/K) :: lhlNI\/~IJK  0 allio. This follows in various ways: either a direct refinement p1'OCI~ "h()win~ that every unit in F is a norm of a unit in K, or from the fact that IIll I (/I,..) . I. 1\Ii Wlili shown in course of proving hZ/l(K*) = n above. Thllll for IInramitll1d I F,
PRELIMINARIES
10
because 'PK/F is the image of 'PL/F under the canonical map C LIF ...... GK/F, and inflation of xa U oX amounts to viewing a character X of C K / F , as a character of GLI F by this same canonical map. Hence
fI2(./F)
=
U
H2(CK/F,KO)
KIF uoramified
the subgroup of the Brauer group H2 (*/ F) consisting of the elements coming from unrarnified layers. We obtain an isomorphism invp: fl2(./F) ..... Q/Z (surjectivity because there exist unramified extension of arbitrary degree). To complete the proof of Axiom II', we must show that the invariant multiplies by the degree IE: F] under restriction from F to E. This follows from (**) when one takes into account that ordE :: e OrdF, where e is the ramification index, and that, under the canonical map C KEjE .... CKIF the image of IfIKE/E is 'P~/F where I is the residue class degree. Hence the invariant multiplies by ef = [E: FJ. This just about completes our introductory comments. Concerning the existence theorem, we have given in Chapter XIV, §6 an abstract discussion which shows that the existence theorem follows in abstracto from Axioms lIlalITe. In both global and local class field theory, these axioms are all trivial to verify except b IIId. The proof of this axiom in the global case is carried out in Chapter VI, §5. In the local case, it is not covered in these notes, but would follow readily from the theory of the norm residue symbol in Kummer fields. Chapter XIII and Chapter XV are not needed for the remaining parts, but note that there is a proof of the principal ideal theorem in Chapter XIIi We hope that the preceding remarks will to some extent reduce the inconvenience which the reader will suffer from the missing portions of the notes, and other imperfections occurring in them.
CHAPTER V
The First Fundamental Inequality 1. Statement of the First Inequality In this entire chapter, k is a globallield and K/k a cyclie cxtCRlliOIl or degree n with Galois Group G. We let J = JK be the idetCli of K, n.nd C = CK be the idhle classes of K. Then G acts on J and 0, and the fixed groupe are JO " Jk, C G =Ok. We let hi and h z denote the orders of the first and soomrl cohnmology groups. ~/l a.bbreviates h2 /h 1 • We wish to determine the order '12 (G'. C) of 'H,2(G, C), and it will be shown in this chapter that h2(G, C) ~ n. In fo.ct, we prove THEOREM 1.
Let k be a global field and let K / k be a qclie ~'c""ion oJ degree n
with group G. Then
or in other fDonLs, h2/ 1 (G,.CK)
= n.
To simplify the notation we omit G and write h.(C) Ill1Iiend ur h,(G, C) whenever G is the group of operators. We shall prove this inequality first in function fields, bccAUHC (lOn..dderable simplifications occur in this special case. Afterward:;, we Hhall Kive .. ullifi,~1 proof for all global fields. We shall make constant use of the pmpertieN of thf! Inrln "'all tlf!vcloped on pages ~7 (Section 3 of "Preliminaries"), and recall here the Lhree RlOit important properties for the convenience of the reader. PROPEIrrY 1. The index ~/l is multipli('.ative. In othr.r wont., If A is aD a.belian group on which G acts, and Ao is a subgroup Invarlllilt undor G we have
~/l(A)
== ~/l(A/Ao)~/I(Ao), in the sense that if two of these quotients are finitfl, then II) II ,he third, IIld the relation holds. PROPERTY
2. H Ao is a finite group, then hall (Au) • I, and hem'e hall (A)
=
"2/1 (AI Ao). PROPERTY
3. If A s;:s Z is infinite cyclic Md r: Optll'IltM trl"'ally, t.hf"n ~/I (Z) ~
n is the order of C.
2. First Inequality In Funrtlnn FINet.
,f"
We rmppose here thFLI. k is " fundiOIl field W" \col U  II" be the unit ideles of K, and Jo ;;: ~ the itlclt:s of volutlll'l I of 1(. 1.11. t~ IIWlIfII 0 IUch that II
u
V. THE FIRST FUNDAMENTAL INEQUALITY
n!ll/I!I'.J1 = 1.
Then Jo J U obviously, and Jo J K" by the product formula.. Hence
J o :J UK·, The muitiplicativity of h2/1 gives
and it will come out that all three quotients on the right are finite.
To begin with, J I Jo is Gisomorphic to the additive group of integers Z with
trivial action under G, via the degree map. Hence
Since the number of divisor classes of degree zero is finite, Jo/U K· is a finite STouP. Hence
1VJ./l (Jo/U K·)
= 1.
The factor group UK· /K" is Gisomorphic to u/(Un KO) and hence
h2 / 1 (UK" /K*)
= h 2/ 1(U/(U n K·» = h2/ 1 (U)(h2/ 1(U n KO)r 1,
Here we use the multiplicativity in reverse, and it will be shown that both ~/l(U) and ~/l (U n K*) are 1. We know that U n K* == is the multiplicative group of the constant field of K, and is finite. Hence h2/1 (U n K*) = 1. We contend finally that h2/I(U) = 1. Indeed, we can express U as a direct pr.oduct,
Ko
U=
IIp (11 U!lI) 'PIp
where each component 11'Plp U'P is semilocal, and invariant under G. For each p let U, be one of the groups Uq!, and let G p be the local group, leaving Up invariant. The 13emilocal theory states that 1(' (G,
II U'P) ~ 1f'(G
p, Up)
!PIp
and we have 1nU);:::
II 'Hr(G
p , Up).
p
We know from the local class field theory that
h1(G p, Up} ::;: h2 (Gp , Up) =
=
e,
where e p hi the ramification index. But ep 1 at almost all p. This shows that h 2 (U) and hI (U) are both equal to ep and therefore that h2/1 (U) = 1, as was to be shown. If we piece together the information just derived, we get the desired result:
11,
3. FIRST INEQUALITY IN GLOBA.L PmLos
13
3. First Inequality in Global Fields , We treat now the two cases simulto.neoWily. The (:xi~tfll1ce of archimedeo.n primes prevents us from giving the same proof for number fields thlLt was given for function fields in the preceding section. Using Ha.a.r me8liUrI', and a generalized Herbrand quotient for Haar measure, one could indeed giw. all argnment in number fields which parallels completely that of function fields. Sill ... k. 1'ht:n thrre exi8t infinitely many primes p that split complete/II in K. (, > 1) lind f"l!maiJi prime in
Kl' PROOF. Let K = K 1 •.• Kr be the compooitum o( 1\11 K". Thf'n K/(K3 ••• Kr) is cyclic. Let q be a prime in (K2 •.. Kr) which r"IIIILiIl~ WillII' ill I\' and which divides a prime 41 of k which is unramified in K. (Thorc elI.lHl IlIllnitl!ly IIIA11y such primes.) Then K q /(K2 ... Kr}q is cyclic of dl!p;rnr p. Uut K~/A:, '" n.lll4l cyclic because p is unramified in K. The Galois !l;roup ()f 1\ IA: IN of tyIX! (P. J'..... p) and that of Kq/kp a cyclic subgroup. Thi~ JIII!IlIlN t.hul II\~ : 1e,1 t; p. Together with [Kq : (K2'" Kr)q] = p this lihoWli (K~ ... K r )4 .. k,. It fullt,wlI tb"t P splits completely in (K2 ... K r ). It remains prime in K 1 , Clr I!Wo 1 E and K/k is normal, NCK C NE/kCE)' In the later case, let S be a finite set of primes including all archimedean primes, all ramified primes, and enough primes such that 1
Jit.
,.*
J:,
= kJg = K* J~. = k* N J~ whence
Then k· N JK == k*· NK" . NJ~
(J" : N h) = (k·
J: :k* N J~) ~ (J: :N J~) ~ nn, PES
by an argument similar to that used in the first inequality (V, 3). This proves the finiteness.
0
The norm index divides a power of the degree because for any a E Ck , aM E
NE/kCE. Let E :J F :J k be two finite extensions. Then: 1. (C" : NF/k(CF» divides (Ck : NE1,,(CE»' 2. (C" : NE/k(CE)) divides (C" : !"fo'lk(CP» . (CF : NE/P(GE»' Consequently if the inequality holds in the steps of a tower, it holds in the tower LEMMA.
itself·
PROOF. We note that
(.)
(C,,: NE/k(CE»
= (Ck : Nplk(CF»(NF//r,(CF): NF/k(NE/F(CE))).
This already proves 1. The map CF .... Nl"/k(CF) is a homomorphism so that the second factor of the right side of (.) divides (Cl" : NE/l"(CE)). This proves 2. 0 From these lemmas we obtain REDUCTION 1. If the second inequality holds in all cyclic extensions of prime degree, then it hoids in aU normal extensions.
PROOF. Let K/k be normal and let i be a prime. Let E be the fixed field of an tSylow subgroup of the Galois group C. K lEis a tower of cyclic fields of degree l and by the lemma we may assume that the inequality holds in K/ E: (CE: NK1E(CK)) divides [K: EJ. On the other hand, (0,,: NElk(CE)) divides a power of [E : k] and is therefore prime to e. From the fact that (Ck : NK/k(CK)) divides (G" : NS/"(C E ») . (CE : NKIE(Og)) it follows now that for each prime f the icontribution to (C" : NK/,,(GK» divides [K : E] and consequently [K : kJ. The inequality for K/k follows. 0 REDUCTION 2. If f # p it suffices to prCJ1)e the second inequality for cyclic fields of prime degree over fields k which contain a primitive loth root of 'Unity, .
IOften in this chapter we write J~ instead of (Jk)S'
3. KUMMER THEORY
21
PROOF. If K/k is cyclic of degree t then the norm index of K/k divides that of K«)/k which in turn divides the product of the one of k(~)/k and the one of K«)/k«). 'The norm index of k«()/k is prime to [since the degree is prime to I.. The norm index of K/k is a power of l and divides therefore the norm index. of K«)/k«), a. cyclic extension of prime degree I. of a field containing (. 0
2. Kwnmer Theory
Let k be any field with any characteristic p, K / k an abelian extension of k (finite or infinite) and G its Galois group with the Kroll topology. Wf' 8hall consider only cases in which G is of finite exponent n, meaning by this that u" = I for allu E G. We shall give an algebraic characteriza.tion of the:ie extension fields K / k in two certain special cases. a. p { n and k contains the primitive nth roots of unity. The nth roots of unity form a mUltiplicative group of order n in k a.nd we UIIe it as value group for the characters of G. If X is a character, its value is in k, so that it is invaria.nt WIder the action of G. This implies x(ur) = x(u)x(r) = X(U){X(TW and shows that the function X(o'} is a continuous lcocycle of (G, K·). 'H,l(G, K4) is trivial. Consequently there exists an 0: E K4 such that X(O') = 0'1 cr. This 0: is not arbitrary, since (0'")1.. (X(I7»n ~ 1 for alIu E G, so that Oi" is in k. This suggests that we introduce the discrete multiplicative !!;roup A of all a E K* such that Oi" E k. We have then X(u) = aia with 0: EO A. If we form converooly with any a E A the function X(u) = Q:l~ then this functiun ill continnous on G since its value is 1 on the subgroup of G that has the field k(a) a..... fix(..'t.i field. It satisfies (x(O'»" = (0:")1" = 1. X(!1) is therefore an nth root of uuity in k and consequently invariant under the action of G. Finally X(U)X(T) = X(O')(X(TW' = ola+",,.,. ~ a l  cr.,. = x(O'r), in other words, X(u) ill a dumll:tcr of C. We introduce now the symbol:
=
(0',0') =
0:1~
a.nd see that it defines a. pairing of the groups A a.nd G into the roots of unity of k. The kernel of A in this pairing, the set of all a E A with a 1 CI 1 for allu E G, is k (k C A is trivial). In order to find the kernel of G let ITo be an clement ;f 1. By the Duality Theorem, there exists a character X !luch I.hltt X(uo) # I. We have lihown above that X(u) = 0: 1 0' for some a. This show:! that Of) i5 lIot in the kernel, in other words that the kernel of G is 1. Let now H be a closed subgroup of G and H.l. the orthogonal group to H under our pairing. Then k· C H.l. C A and any subgroup of A tl1"t mlllll.llll! k will come from precisely one closed subgroup H of G namC'ly its ortbogonwlI;roup. We adjoin the elements of H· to k and obtain the subfiekl k(1I J) of /\' Whil'h tmugl'Oup of G determines this subfield? It consists of thOti(~ elmntllltH t1 of a which leave every element of II ~ invariant. The translation of this ~tulClIl('lIt. illto the IIlI1K\I~e of the pairing shows thl\.t the group in question ill orth0l1.0uw to H J. Anmding 1.0 the duality thL'Ory it ill therefon' /I. We have tht'!r('foct'! 1\ I 1 corrctlp4.llld{~u{:e bctween the subfields of K Ik a.nd the !lubgrollps of A t.hllt contain k". The oonnAct.ion between the structure of G and that of A ill &Ivt1n by
=
A/k" ~G
VJ. SECOND FUNDAMENTAL INEQUALITY
22
~
and the map defined by our pairing.
The group A can now be mapped onto a subgroup of k by raising each element into the nth power. The kernel of the ma.p 0 ..... 0" consists of the nth roots of unity and is therefore contained in the previous kernel k. All our statements can therefore be compounded with this further map. We obtain A"/kn ~ and have a 11 correspondence between the subfields of K/k and the subgroups of An of k" which contain k ..'. We must now do these steps in the reverse order. Suppose we start with an arbitrary subgroup ~ of k· which contains k*n (this ~ is to play the role of the final group An). We extract all nth roots of elements of ~ 8Jld obtain a group ~l/n in the algebraic closure of k. (~l/ .. is to play the role of A. It clearly contains k.) Construct the field K = k(~l/n). It is a normal extension because it is a splitting field of the separable polynomials x"  6 of all J E~. Let G be its group. We contend that G is abelian. It suffices to show that the action of G on each generator Q E ~l/n of the field K/k is commutative. We have 0" E k or (a")O' "" an for each U E G. This shows that aO' = (",0 where (" is an nth root of unity. If.,. in G, then 7' leaves (0' fixed and we obtain (0"')" = ("(.1'0, an equation, which shows the required commutativity. We also see u" = 1 whlch shows that the field K / k is of our previous type. We determine now the group A in this field K. Clearly A:::) Il i / .. :::) Ie*. Since subgroups of A are in 11 correspondence with subfields of K and since k(lll/n) "" K it follows, that actually A = ~lln. We summarize our maio results in:
a,
THEOREM 3. Let k be a field and suppose that the primitive nth roots of unit !I lie in Ie, p t n. There exists a 11 corre.spontknce between subgroups ~ of k" which contain k .. n and abelian extensions K/Ie with groups G of exponent n. The
coTTeSpondence is given b!l ~ ++
KA
= k(~l/n).
The character group {J is isomorphic to the factor group ~/k"n. The following corollaries are obvious: COROLLAR.Y 1.
1f ~1' ~2 are two subgroups of k" containing k*" then we have
the oorrespondences Al n~2 ..... K A • nKAa' COROLLARY
2. If K/k is finite then G ~ 8 and [K : k)
= (.:.\. : k*").
~
:::) k*". We now remark that if .:.\. is a subgroup of k' which does not contain Ie''', we can always form the composite group ~. Ie·". Then the Kummer group of the field K "" k(~l/") is ~. k·". The factor group .:.\.. k·"/k·" is isomorphic to ~/(.:.\. n k*") and in case K/k is finite, we get [K: Ie) = (~ : (.:.\. n k·"».
In all our discussion, we have taken
COROLLARY
A/k*n then K
3. K/k is cyclic if and only if ~/k· is cyclic. If 0 generates
= k(V'J).
FOI global fields these statements have to be supplemented by arithmetical .atements:
2. KUMMER THflORY
4. Let k be a global field containing the primitive nth roots oj 'Unity, Let K = k(.1..1/n) be a finite Kummer extension and, a prime oj k. 1. P splits completely in K if and orny if.1.. C k;". 2. Suppose that p is finite and that p n. p is unram(fied in K if and only there exists a set oj generators of .1../k" which are unit" at p. In other words: THEOREM
Pf n. if
t
A C Up
·k·".
PROOF. 1. The complete splitting of p means that the completion of K at a prime div:iding pis kp• Since this completion is kp(.1.. 1 / n ) the contention is obvious. 2. Let I:j3 I p in k. K x = o. We fonn the field K = k(~A) and arrive by completely simil&.1' arguments at the theorem:
.
.
5. Let k be a field oJ characteristic p > O. There is a 11 COm!qondence between the additive subgroups A of k which contain pk, and abelian fZtensions K (k with groups of exponent p. The correspondence is given by THEOREM
A K~ = k (~A) . S. Proof in Kummer Fields of Prime Degree
Let k be a global field, of characteristic p (= 0 or > 0). Let n be an integer, p f n. (n denotes a primitive nth root of uwty. We assume ( .. lies in k. We shall introduce auxiliary groups of ideIes in the following way. Let S be 8. finite set of primes containing at least all archimedean primes and divisors of n. This set S is split up into two disjoint sets 81 and 82, one of which may be empty: S = 81 US2. Let i stand for one of the two subscripts 1,2 and call j the other. We define the group Dj to be the group of all ideles a such that 1.
11
is an nth power at all P E Sj,
2. a is a WLit outside 8, whereas no condition is put on the primes of Sj. We see that Di can be written naturally as a product:
D. =
II k;n x IT k; x IT Up. PES,
pES,
p¢s
n
=
An element a E Di can be written a b" . ( with b E ll.ESJ k;, and C E pE$. k; x npl!sUp, We put Ai = Di n k. It is clear that ks J Ai :J k;;" and we know from the Unit Theorem that k'S/k'S" is finite. In fact; k'S is a free abelian group on 8  1 generators if S consists of s primes, modulo the roots of unity. Since k contains the roots of unity form a cyclic group of an order divisible by n and this shows
e.. ,
(1)
(ks: k'Sn)
=nS.
The extension Ki = k(AVn) is a finite Kununer extension, belonging to ~ik·". We obviously have.A i n k· n ksn. From this we get for the degree
=
[Ki: k]
= CAik"" : kon ) = (Ai: ksn),
by Theorem 3, Corollary 2. In addition, from Theorem 4 we obtain 8. crude deIICI'iption of the splitting of primes in Ki as follows: If p r/:. S, then p is unramified in K i . If pES;, then p splits completely in Ki . In a similar way we define Kj. We denote the ideles of Ki and K j by J. resp. JJ and norms from these fields to k by Ni resp. Nj. The proof of the second inequality is contained in the following lemmas. LEMMA 1.
Let K(k be a finite abelian extension 0/ exponent n (i.e. un Then on E k· NKI"JK'
.,'17 E G). Let a E J".
= 1 for
3. PROOF IN XUMMD FIELDS ;;, PlUME DEGREE
:18
PROOF. We first note that even though G has exponent'n, it does not mean that G is of order n. If G has order n, then the lemma. is of course trivial. If we wish to use local class field theory, we can even prove tha.t Q" E N J K. Indeed, Q p E NpK;, SO the lemma is obvious. We can give an clcmentl\1'Y proof, however. Let a E k* be such that aa is close to 1 at all ramified primes. Then (0'4)" is norm at these primes. 2 At the other primes, the local extension is cyclic, and therefore has a degree dividing n. Hence (oa)" is a norm at the unramified primes also. This proves the lemma. 0 The next lemma is fundamental, and shows bow the Kummer theory of the two
fields K t interminglC5 witb their class field theory. LEMMA 2. Let k contain the nth roots a/unitt!, p f n. Let S be a finite set primes including • all pIn and all archimedean primes, • enough primes 80 that Jk = k"
0/
Jr,
Let 8
= 8 1 U S2
as above. Let K, = k(.:1V rt ). Then 1. k"DiCkONjJj. 2. (Jk : k* Dl)(J,. : k' D 2) =: [Kl : kJ[K2 : k) and consequently
(J" : kO NIJ] }(J" : k* N 2 J2 )
~
[Kl : kJ[K2 : kJ.
PROOF. We begin with the first statement. Let a E D j • Write a == b" . ( as above. Then b" E k·NjJj by Lemma 1. ~ bas (:omponent 1 at all " E 8 j • FUrthermore, if P E Sj then p splits completely in K J , and hence tp is a local norm. If p ¢ S, P is unramified, tp is a. unit and consequently a local I1Hrm. Thus cp is local norm at all primes, and hence t E NjJj. This proVP.S that a€; k" NjJj • The second statement will follow from a brief inll(~x computation. We liM remind the reader of the general rule: (A; B) (AC : BC)(A n C) ; (B n CJ). We ha.ve tberefore
=
J: :
(Jk: k* D i ) == (ko. k" . D;) _ (Jf: D,)  ({Jf n k') : (D. n k*») (Jr: Di) = (k .:1.) k" . kO") ." PES, II' II (a. kO") (k k j. s kO . k") == PES, P' II [Kj : kl. n'
s: n (
s: s")
n (
We form now the analogous expression for (JI< : kDj ) and wultiply the two
formulas. This yields (J,. : k" Dl)(J" : kO D2)
'=
nPES:' (k"' k*n) " [Kl: klfK2 : k]. n"
2See Ch. 11. Sections" and 5. for the deta118 of Jun haw ~a. to 1 ("')" n~. to be. But ,he fact that NpKp on is open i. more elemental')'. For cxampl0. iL col\l.alM k.1 whid! ia compact of finite index in Up. hence open
"t. ·
'') VI. SECOND FUNDAMENTAL INEQUALITY
In the remarks on pages 6 and 7 we computed the local index
(k, : k;) Since
= n2/lnlp·
Inlp = 1 for all p ¢. 8 we get from the product formula
nInlp =II Inlp =
PES
1.
,
and consequently
IT {k; : k;") =n
2•
PES
which leads to the de5ired fonnula
The second inequality will now be an immediate consequence of the next lemma. Note how the Kummer theory relates Ki with D; whereas the class field theory relate5 K; with Dj . 0 LEMMA 3. Let n be a prime. Let k(v'ii) = K be any given Kummer extension ·oj degree n. Then there exist two disjoint sets of primes 81 and S;. whose union S satisfies the condition of Lemma 2, for which J" = kD1, and lor which the GSsociated field KI happens to be K.
Before proving this lemma we indicate right away how the second inequality follows from it. Since J" k" DJ C k· . N2J2 we have (J" : k" . N2 J2 ) 1. By ·the first inequality this implie5 that K2 = k. The last line of Lemma 2 reads now (Jk : k* NlJI) ~ [Kl : k] and this is the !IeCOnd inequality since K = K I . (The divisibility follows from the first lemma of this chapter.)
=
PROOF OF LEMMA 3.
=
Let 8 1 be a finite set of $1 prlme5 containing
• all pin and all archim.edean primes, • all P I a (which makes a an 81unit), • enough primes so as to have Jre = kJt' .
=
To simplify the notation we put J = JII, and JS1 J:1. ksJks, is of type (n, n, ... , n) since n is a prime. Q is not an nth power and · may therefore be extended to a basis of kS11 k~I' Let al •... ,as, be such a basis, Q=a1'
Put EfA. = k( va,;). Then K = E 1 • The full compositum of these fields has
d~
sree n"', the compositum of all but one field nB,l. This follows from the preceding eer.tion. Our fields satisfy therefore the condition of Theorem 4 of Chapter V so that there exist primes II. outside the set 8 1 such that IIi remains prime in the field J::. but splits completely in all the other fields EjJ' Of these primes we utilize only the prime5 112,113, .•. , q., and form with them the set 82 (so without the prime 111)' Then al is &Il nth power at all the primes of 82 and ai (for i '" 1) is an nth power at. any qj I IIi but not an nth power at IIi.
C. PROOF IN ,..EXTI::NSIONS
We look now at the structure of the groups J 51'
Dl and J s. n D 1 :
",
Js, ==
n nU n k; x
pES,
q,
x
.:2
Up,
pf/;S
",
Dl ==
IT k; x IT k·~. x 11 Up. pES,
Js, n Dl ==
.=2
pf/;S
••
n kp x nu:. x n
PES,
i~2
Up.
pfl;S
The local index (Up : Up) has been computed on p. 7. Its value is n/lnl,. For a prime outaide SI this is simply n since all pin are in 8 1, n iN a prime and so the factor groups UqjU;' are cyclic of order n. If we can find idcles a, E Js, which are nth powers at each aj # Ilj and not an nth power at Il; then these ide1es Ilj will generate the factor group J5.1 JSI n D l . The peculiar properties of a3, ... ,a" show that these principal idcles (they are in ks t ) will serve. Since they are principal idelesweseethat JS l c k*·(Js,nVd c k*Dl andoonseqllent.ly J = k·Jsl C kOVI which proves the first part of our lemma. We have still to show that K Kl = k(Ajln). We remark to this effect that the group VI (see the display of its structure above) i~ obvioUBly a subgroup of Js,' In :: Js 1 (k'JS1 )R = k*nJs,. A generator Ii of the Kummer group Alk'" mod k·" of the field Kl can therefore be assumed ill Js1 • i.e. in ks •. It has then the form Ii a~ ... a:~1 .
=
= art
But Ii has to lie in A1kn which implies that 6 must he an nth power at every prime qi E 8 2 (i 'Ie 1). At qi every Uj with j 'Ie i is an nth power and a; is not. This shows that 112,113,"" II" are divisible by n. Since nth powerll ('an be absorbed into k" we ma.y now assume 5 = ai'l. Such a 6 is indtlt!d ill Al 5ince it is an nth power at every prime of S2 and a IlD.it outside S. Since UI Q this proves the contention. 0
=
4. Proof in
~xtension8
a. A lemma on derivations. In the course of the proof B certain lemma on derivations will be needed and we prove it separately becRlllw. it is of independent interest. By a derivation D of a field E one means an additive map of E into itself which satisfies the usual rule D(xy) Putting x
= :rD(y) + yD(z).
=11 = 1 one obtains D(l) = O.
Letting now II
= xlODe _
D(:r 1) == x 2 V(x). The additive group of all x E E for which V"(J:) = 0 eha.ll be denoted by P". = identity, hence Po "" {o}.) As usual, we denote the additive operator that multiplies the clements of E by a given element 11 of E also by y. A confUSion hfltWI1('1l II l1li Ilif'nlllnt and 11 as operator is avoided by the use of parelithCliiN: Vy III1!B/1I> Lilt' optlralor product II followed by V whereas D(y) means the effcct IJ hAIl 1m thn nlftmnnt 1/.
(1)0
•
VI. SECOND FUNDAMENTAL INEQUALITY
We say that an element x E E ill an element y E E" such that x = derivative of y.
8.
loga.rithmic derivative in E if there exists We say then that x is the logarithmic
¥.
LEMMA 4. An element x of E is the logarithmic derivatiwe of an element 11 E P,. which is not in P"l (n > 0) if and only if the nth power 01 the operator D + x applied to 1 is 0 and the (n  I}st power applied to 1 is different from 0: (D + x)"(I) = 0, (D + x),,1(1) #: O. PROOF.
D(z)
¥
Let:z: = for some y E E'". For aU Z E E we have (D+:z:)(z) "" =: 1I1D(yz) = y1Dy(z). This means in operators that D + x =
+ ¥z
,,1 Dy and consequently for any n ;;.: 0 that (D + x)"
=:
yl D"y.
Assume now in addition that 11 E P" and 11 f/. P,,l. Then (D + x)"{I) = D"y(l) = y1 D"(y) = 0 and similarly (D + x) ,,1 (I) '" O. Assume conversely that (D + x)"(I) = 0 and (D + x)nl{l) '" O. Put (D + x)"I(I) = 1/1.
,,1
= ¥.
Then (D+ x)(y1) = 0 = _y2 D(y) +xy1 and consequently x Therefore D +x yl Dy, (D + x),,l(l) =: y1 DOl(y) '" 0 and similarly 1/1 Dn(y) := o. Hence 11 E Pn but 1/ f/. P,,l' 0
=
Frequently the derivation D has the additional property that for ewry x in E 0 for some n or in other words that E is the union of all Pn . Then our lemma shows that an element x E E is a. logarithmic derivative in E if and only if (D + x)"(I) =: 0 for some n. If x is now an element of a. 6ubfield F of E tha.t is stable under D, (D(F) c F), and if x is an element of F. then x is a loga.rithmic derivative in F if and only if it is a logarithmic derivative. in E. This situation occurs in the followiug case: Let k be a field of cha.racteristic , > 0 and E k{t} the field of formal power series in t with coefficients in k. If D is the ordinary differentiation with respect to t then DP = O. We obtain therefore
we have Vn(x)
=
=
COROLLARY. Let E = k«t») be a power series field of characteristic p > 0 and F a subfield stable under the ordinary derivation D "" 1;,. An element x E F is logarithmic derivative in F if and only if it is a logarithmic derivative in E.
b. A pairing connected with function fields. Notations: k a global function field of characteristic p > O. ko its constant field, Zp the prime field. kp the completion of k at a prime p and also the additive group of k". the multiplicative group of kp. k} the residue class field of kp • k} contains naturally ko, and k" may be viewed naturally as the power series field k 1 where t is a local uniformizing parameter, i.e. an element of ordinal 1. If necessary, we can select t from k and even assume tha.t it is separating, i.e. tltat k is separable over ko(t). (Every p is regular in the sense of 12, XVII, 4) becauae ko is perfect.} S the trace from kl to Zp. res(x dy) the residue of a local differential x dy computed with any local uniformizing parameter (it does not depend on the choice of the parameter). The reader will find the necessary explanations and proofs in [2, X, 3].
kp
«t»,
.co. PROOF IN ,..EX'l'ENSIONS Q.
A local pairing. For x and y in
J,
(1)
xdy
k~
we write
"
= S(res x dU);
If x E k, and 11 E k; we define 'P1I{x,y)
Lx Y Lx ~
=
dy =
II
p
dt.
11
We find easily !pp(x + x', y) = 'Pp(X,lI) + 'Pp(X', y), '(e» = L 1{p cit. ~
II
LEMMA tl.
If
eis a valuation vector such that I exdt = S'(>.({x» = 0 for all
z E k th.en { is in k.
QX for any Q E ko and obtain S'(Q'>'({x)) = O. IT were '# 0 we would have S'(ko) = 0 which is not true since leo/'Ll' is separable. Therefore >.({x) 0 for all x E k. Theorems 5 and 9 of [2, XIII, 51 show now that (Ek 0
PROOF. We may replace x by
'>'(~x)
=
t is separating in Ie. From [2, XVII, 4) it follows that for almoot all p the derivatives d~: are local units (t, means a local Wliformizing parameter at the prime p). If Q E J" is an idele then its local components ap are in If we form the vector with the components
k;.
1 dap
1 dap ( dt dtp
;; dt = Op at"
*.
)1
theIl almost all of its components are local integers, it is therefore a valuation vector that we shall denote by ~ We define now the following pairing between elements x E k and ideles a, into Zp:
(2)
'P(z,a)
= fx~: cit = Llx~ cl~ dt = a
"pOp
Lix
clOp
j l p Qp
= L~p(x,Op). P
The last expression shows that we have indeed a pairing. As for continuity, nothing has to be shown for the discrete k; if Op is very close to 1 then ~ is very close to 0 and a look at the components of the local integrancis shows the required continuity on J". The moot interesting question, that of the kernels is answered by LEMMA 1. The kernel of J" in our pairing (2) is precisely k· '.If, that of k is the additive group ph, i.e. th.e elements of the form zP  Z for z E k. (Note that every element of k has period p and that J,,/k' . .If is compact; indeed it is
isomorphic to
Ck/cr,.
The duality theory may therefore be used.)
Hr
PROOF. 1. Suppose Q is in the kernel. The valuation vector ~ ... has then the property that § ~x dt = 0 for all x E k and is therefore an elemeIlt y of k. Taking the p component of y at the prime P, which has the element t E k as local uniformiziug parameter we obtain
ldap
r=· lip dt '!I is therefore a logarithmic derivative in kp • The field k is a BUbfield of kp and is stable under the differentiation since t is in k. '!I is also in this subfield and the corollary of Lemma 4 shows that y is a logarithInic derivative in k:
1 dz
r=;dt'
ZE
Ie.
al
Co PROOF IN pEXTENSIONS
=
We obtain for our ideIe: H~ == ~~. Put c = ,,III; we find easily ~ _Z2~1I+ O. Each component satisfies therefore ~ = O. This means that each component i5 a pth power whence C E J: and consequently a E k' . Jf. Both Jf and k lie trivially in the kernel, k for the reason that>. is a differential. 2. Suppose x = zP  z. By the approximation theorem we can find an element 11 E k such that the idele yo is very close to 1 at all priml'B where x has a pole. At these primes then \Pp(x, yap) = O. Furtheremore field to k' D2' It is then no wonder that the existence theorem comes out as a byproduct.) Let a E Ck be universal norm. It is in particular norm from K 1 . It has therefore a representative ideIe a E D2 • Any two such representatives differ by an clement 6 E .12 k~. a is nth power at all primes in S. We shall prove that a is al80 an nth power at all primes outside S. If S/ :> S then a has a representative & e D!i C D,. We can write b ,nb/, where hi is unit outside S' and is 1 in S', and , is 1 outside S'. Let {3 E k be such that c {3" where (J E Js. Then ,n {3nf)" and b = {3nt}n b' . This shows that b is nth power at all primes in S'. b differs from a by IIJl nth power and therefore " is also nth power in S'. The set S' can be lWIde to include any prime, and this concludes the proof of the theorem. 0
=
=
=
=
6. Sketch of the Analytic Proof of the Second Inequality
The Kummer theory proof in Section 3 is basically Chevalley's (cf. [6, Sect. 9)). Before 1940, the only known proof of the second inequality was analytic. The analytic proof is quite short, once given the behavior of thp zeta functions and Lfunctions near s = 1. It works for arbitrary finite extensions K/k, not necessarily even Galois, but gives only the inequality, not the diviBibility. The analytic part of the argument, after the properties of the ( aud L functions are established, is the same as the one Dirichlet used to prove his theorem on primes in arithmetic progressions. He showed that for a prime to m, the set of primes congruent to a (mod m) has what we now call "Dirichlet density" Ijtp(m). A ~t T of prime ideals of k has Dirichlet density 6, if the function "" _1__ 6 Iog(_l_) L.JTNp· s1
pE
is bounded on 1 < IJ < 1 + E. The set of all primes has density 1. This follows from the fact tha.t the zetafunction (k(S) = fIll (1  NP·) 1 has a. simple pole at s = 1, so that log (k(S) log (.':1) is bounded uear 8 1, and on the other hand, log(k(S) differs from I:, by a bounded amount there. A finite set, and any set of primes of degree> 1, have density O. The analytic proof that II == ICk/NCKI is less than n = \K : k) is by showing that the set of primes p of k whicll split completely in K has density ~, and the set of primes whose class is a norm from K has density Since the former set is contained in the latter this implies h ::; n. A sketch of how this goes is a.s follows. Let B be an oprn subgroup of finite index in Ck' As ClCplained in (V1II, 2 and 4), we can view C k / B as a group C of classes of fractional ideals prime to a "conductor" m, each CUI:lS being a. coset of a group of "generalized arithmetic progressions" (mon m). Let X denote the character group of C, the analog of a finite group of Dirkhlet characterlJ. The as:>ociated Lfunctions, defined for Re(s) > 1 by L(II, X) = n,,~ (I  w.)1 have similar properties to those of Dirichlet Lfunctions. They are holOinorphic at s = 1, ClCcept for X 1. The zeta function, (k(S) L(s, 1), haa a pole of order 1 there.
*"
=
*.
=
=
VI. SECOND FUNDAMENTAL INEQUALITY
One concludes, following Dirichlet, that
2: ~~; = ax log C~ 1) +0(1), p,/ln
for s near 1, where
ax Is the order of L(s,X) at s = 1.
finds
L X(p)=l
=
1 N ,
P
Averaging over X E
X one
= ~(1 Lax)logC~l) +0(1), x;tl
where h = IXI lei = le,,1 HI. . Now suppose Klk ~ a finite extension, and B = NK1"CK, so that h Is the Dorm index. Let T be the set of primes p of k which split completely in K. Each PET lies below n = [K: k] primes ';p of K, each with N';p = Np. Hence, since the set of all primes ';p of K has density 1,
1 = n12: N';p' 1 S 1n ,L..J,N!ps 1 = log 1 (1) + 0(1). L Np' n s 1
pET
\lilT
..U\lI
The primes PET are norms from K, so their claases are norms, i.e. they are among the primes p with x{p) 1. Comparing the last two equations we find therefore
=
~ S i(l Lax)' x#
Conclusion: ax
= 0 for X::F 1, i.e.
L(s,X) does Dot vanish at s = 1, and h S n.
CHAPTER VII
Reciprocity Law 1. Introduction
Ie is a global field, n the separable pwt of its algebraic closure, and 15 the Galois group of fl/k. Gn are the idele classes of fl, Gn UK GK where k eKe fl, K/k finite and normal. We have seen in the preceding chapter that (15, Gn ) is a field formation, i.e. 'Hl (~, Gn) = 1 for all open subgroups ~ of 15. It is possible to assign invariants to the element of 'H2 (15 , Go) in such a way that (15, Go) becomes a class formation in the sense of Chapter XIV. This assignment will be carried out in this chapter. In function fields, the situation is very much like that in local fields: every idllie class can be given an ordina~ and the constant field extensions in the large can be used in the same way as the unramified extensiollS were utled in local class field theory. In number fields, it turns out that thc cyclotomic extensions can be made to play 8 role similar to the constant field extensiollB, in spite of the fact that they ramify and that they are not all cyclic. (As a matter of fact, the constant field extensions themselves are cyclotomic, i.e. they are obtained by adjoining roots of unity.) It win therefore be necessary to prove first the reciprocity law for cyclotomiC extensions of the rational numbers. This is done in 2. The proofs proceed in a completely elementary fashion, except at the very last argument where the second inequality is used. From then on the reciprocity law can be proved for normal extensions K/k of global fields without cssentia.lly distinguishing the two cases. We first assign invariants to idele cocycJes by taking the sum of their local invariants. We then prove that number cocycJes have invariant O. Thill is done by moving a number cocycle from a normal extension to a cyclic cyclotomiC extension, where this fact has already been proved. We select the auxiliary cyclic extcllBion so that it splits the cocycle locally everywhere, and then use the triviality of 1l 1 (GK) to move the cocycle. By this procedure, we get invariants for idele class cocycles, whenever these have a representative idele cocycle. When they do not, the triviality of 1l 1 (GK) allows us to move a cocycle to a cyclic extension K' on which it splits. In K' the cocycle has a representative idllle cocycle, and hence may be given an invariant there. It is easy to show that the invariants thus obtained &.rI: independent of the auxiliary constructions performed, and that they satisfy the axioms of a class formation. Knowing that (e,en) is a class formation, we can apply the axiomatic devel· opment of Chapter XN. In particular, we get the triviality of the third cobomology group 1l3 (G, GK) in finite layers of the formation.
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VIt. RECIPROCITY LAW
We also get the existence of a hOIDomorphism w of q. into a dense subgroup of f}/0' given by the norm residue symbol. In function fields, the situation is again oompletely analogous to that of local class field theory. w is an isomorphism, but is only into f}/0'. It is possible to complete Ck to a group ak in such a way that w extends to an isomorphism of k onto 0/6', by a procedure similar to the one carried out in the local class field theory. In number fieldB, the situation is different: The norm residue symbol w map5 Ck onto 0/6', but there exists a nontrivial kernel, the elements which are infinitely divisible. This kernel turns out to be the connected component of Ck, whose structure is discussed in detail in a later chapter.
a
2. Reciprocity Law over the Rationals
Let Q be the rationals, Qp the completion of Q at P and Poe the archimedean prime of Q. Then Qpoc is the reals and contains the multiplicative group R+ of the positive reals. V denotes as before the product of all Up for p #: POOl considered all subgroup of the ideles JQ. JQ contain.ion obtained by adjunction of all roots of unity shall be called the maximal cyclotomic extension of k and any intermediate field a cyclotomic extension of k. An automorphism of the maximal cyclotomic extension hi described by its action on the roots of unity. A root of unity is sent into a power of itself and one derives easily from this that the Galois group is abelian. Denote by the maximal cyclotomic extension of Q and by p the maximal cyclotomic extension of Qp. rp contains a maximal cyclotomic extension of Q as subfield and we have r p = fQp. is isomorphic to r but there are many isomorphisms of onto r. If t is one of them then At is the most general one, where .x is any element of the Galois group of r /0.. Let 6 be the Galois group of r/Q and ~ the one of f/Q. The ma.p t induces a natural map i'i = t1ut of ~ onto~. Since 0 is abelian one sees that this map is independent of the choice of the map t. This allows a simplified description. We shall choose an identification of the field with the field r, so that we may say r" = ro.p. This identification results in an identification of (5 and ~ and this latter identification is independent of the way in which the identification of rand r is done. The Galois group 0" of r p = ro.p over Qp is now mapped in the well known way canonically into 0, by looking at the effect that an element of ~p has on r. The group ~p becomes in this way a closed subgroup of 0. For p == Poo the field I'Qp is the field of complex numbers and ~,. a group of order 2. If p is finite, rQp/o. contains the maximal unramified extension ofQ,., obtained by adjoining to it all moth roots of unity (0 with a.n mo prime to p. The group of
with the product topology (Q* discrete) since R+
r
r
r
r
r
r
2. RECIPROCITY LAW OVER THE RATIONALS
this unramified extension contains the Frobenius substitution which sends each (0
onto (C. We shall denote by'Pp an elernent of t,5p which has this effect on the (0. Our first aim is to prove tha.t 6 is isomorphic to U in a' natW'al way. An element of 0 is determined by its effect on the roots of unity. If ( is a primitive mth root of unity then the automorphisms of 0 induce of course the Galois group of Q()/Q OD this subfield. The following lemma which is nothing else than the irreducibility of the cyclotomic equation givC:i the structure of this group: LEMMA 1. Let ( be a primitive m·th root of unity. An automorphism oIQ(t;)/Q sends t; into a power ('" where n is prime to m. Conversely, to any given n prime to m there is an automorphism a such that (a == t;". In short, GQ({}/Q Rl (Z/mZ)·.
PROOF. The first part of the lemma is triviaJ.. AJ3 to the second part, it suffices to prove the statement if n is a prime p that does not divide m. (satisfies the equation xm  1 = f(x) = 0, and f'«) = m(ml is prime to p. The local field Qp()/Qp is therefore unramified. Its Frobenius substitution sends ( into an mth root of unity that is congruent to (P. Since r(O = 0,.((  (,.) is prime to p it follows, that no two mth roots of unity are in the saDIe residue class; (P is therefore the image of ( under the Frobenius substitution. This automorphism of the local field induces an automorphism of the global field Q«)/Q and this proves the wmma. 0
For the description of the automorphisms of the infinite field r the exponentiation with integers is not convenient and shall be replaced by an exponentiation with elements u E U. Let mp be the pcontributions to m of the prime divisors p of m and up the pcomponents of 'U. We can find an integer n that satisfies the simultaneous congruence n == Up (mod mp). We write shortly n == u (mod In). This n will be prime to m and its residue class mod m is uniquely determined. If we put ('" = (n then ('" is well defined. If conversely an integer n prime to p is given, there exists a u E U liuch that u ;: n (mod m). Indeed, it suffices to select Up = n for all 11 I m and Up "" 1 for all other p. One verifies easily tha.t «U)tI = (Uti for 'U and v in U. According to oW' lemma we may now say that ILIl automorphism of Q«)/Q sends ( into some power (" and that each u E U giVC5 rise to an automorphism of this field which maps ( onto (". If m divides m' and if (' is an m'th root of unity then Q«) C Q{('). Let a be the automorphism that sends (' into «(')11.. To describe it we have to find an n' == u (mod m'). Then «(')0 «,)n'. ( is a power of (' hence (" = (n'. Since this n' is also congruent to u (mod m), (" = (u. This means that the automorphism that u defines on Q«') will induce on the subfield Q«() again the automorphism corresponding to u. Let now U u be the following map of r into r. If Q E r then Q Iiee in some field Q«). The given u defines on Q«) a certain automorphism and we let a .. (Q) be the image of a under this automorphism of Q«). If Q lie~ aJoo in the field Q«'), we can find a root of unity (1 such that ( and (' are powers of (1. The automorphism of Q«l) corresponding to u agrees with those of Q«) and Q(') I\lId this ::;hows that our map is well defined. au is an automorphism of r sin(:e it ilIlI.ll automorphism 011 every sub field Q((). To make its description now very short; au liCllW; every root of unity, into 1'r
_ ..up, ,>p'
(!{)p
Frobenius subst.)
The map O'p ~ up(op) is continuous. The kernel is R+ at p"" and 1 at finite 1
primes. Indeed, to have up = 1 we must have (;, == 1 for all (p. and this implies v;l ;;: 1 (mod prj for all r, whence Up I. It also implies !()~rd(O:p) 1 and this
=
=
means ord(O',,) = o. The symbol u1'(O'p ) will eventually tum out to take its values in 6 p and to he the local norm residue symbol of Qp. We ha.ve obviously: THEOREM 1. If u p (lI) for up(ap).
a has components
lip
then
0'(0) =
11" Up (ap).
We also write
We come now to the keystone of this investigation: THEOREM 2. Let K/Q be a finite cyclotomic extension: Q eKe Q«) where Let ~ be the subgroup of f!3 that dete7"l1l.tnes K (is identitu on K). If a is id~le E Q. NKIQJK then 0'(11) E ~.
('" = 1. an
PROOF. 1. For each p the map a p + CTp(ap) is continuous and ~ is open. Therefore O'p(O'p ) E ~ for all O'p near enough to 1. 2. Let S be the set of all a.rchimedean primes of K and of all primes that divide m. Assume a = oNK/Q(~) with 0' E Q. According to the approximation theorem we can find an element A of K such that 2l = AlB where !B iii as close to 1 as we like
VII. RECIPROCITY LAW
at all primes of S. Then 21. = ONKIQ(A)NK/Q(~) ;;;: fJNKfQ(I]j). Since u(fJ) = 1 it suffices to show a(NK/Q{IJ3» E I]j. We decompose according to Theorem 1 this automorphism into its local parts C1p (NK/Q(IJ3» up" We prove the contention by showing that each Up is in ~ (~ is closed). The norm mapping is continuous. If we have brought the components of I]j for the primes of S near enough to 1 then NK/Q(IJ3) will have components near 1 for Poo and for all p I m. According to what we have seen, up will be in ~ for these primes. There remains the case when p is finite and does not divide m. The field Qp«() is then unramified and so is the completion Kp of K as subficld of Qp«). If np is the degree of K~/Qp and 'lJp the F'robenius substitution then rp~P leaves Kp and consequently K C K~ fixed. The pcomponent of N K /Q (I]j) is a certain local norm N K p /Q. (a) and all we have to know about it is that its ordinal is a certain multiple rnp of the local degree of Kp. Since p does not divide m, the action of up on ( is the same as that of rp;np. The automorphism uprp;rnp leaves therefore Q«() and consequently the subficld K fixed. Since rp;np leaves it also fixed it follows, that K is fixed under up in other words up E ~. 0
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We remark that Theorem 2 contains an independent proof of the first inequality bI. cyclotomic extensions of the rationals. The local counterpart of Theorem 2, together with the proof that the map op ...... up(Qp ) is a true local map is contained in:
Q;
THEOREM 3. The map a p + up(Op) is a continuaus homamorphism 0/ onto an everywhere dense subgroup 0/6 p. Let Kp be a finite cyclotomic exten6ion 01 Qp and ii the (closed) subgroup of 15 p that deteffllines Kp. If 01' E is a norm from K~ then 0"1'(01') E ii.
Q;
PROOF.!. We begin with a proof of the last part of the theorem. Asrrume that 01' is a norm from Kp and let I1p be the idele with component 01' at p and components 1 at all other primes. Let ~ be any open subgroup of the global group (it has finite index in 6). The group ~~ is also open and determines a certain finite extension E of Q. The completion of E at a prime dividing p is determined by the subgroup ~ii n I5p of 151" Since ij is in 15p we have i; C ~i) n I5p. This means that the completion of E is a 5ubfield of Kp so that a p is also a norm from the completion of E. The idele IIp is consequently a norm from JE and Theorem 2 shows now O"(ap) E ~ii. u(ap) is therefore in every open neighborhood of the compact, hence closed subgroup ~ and consequently in ii. Hence O"p(ap) E ii. 2. If we select Kp = Qp. then ii = 6 p and every O'p is norm from Kp. Therefore O"p(ap) E 61" Our map is therefore into 6 p • 3. The image of will be everywhere dense in 61' if every automorphism of a finite extension Qp«)/Qp is induced by some up(op). If we write = (o(p~. where (0 is an moth root of unity (with p I'l'10) and (pr a prth root of unity, then Qp(Co) is unramified. The action of an automorphism on (0 is therefore the same as that of ~ where rpp is the Frobenius substitution. (pr is sent into say C;:" where (n,p) = 1. According to Definition 2 the element O"p(n1pi') has the same action. 0
o
Q;
t
nvp(Vq'1) = 2: invq cq. "Ip
'Th.king the sum over p proves the lemma.
qljl
o
VII. RECIPROCITY LAW
Our purpose is nOW to determine the kernel of the homomorphism c i.e. to characterize the elements c E 'H.2(G, h) which have invariant O. To simplify the notation we omit G in writing cohomology groups. We recall the exact sequence
1(l(CK)
+ 1f(K*) +
1f.2(h)
0
+
illv c,
'}{.2{CK).
It was a consequence of the first two inequalities (VI, 1, Theorem 1) that '}{.l(CK ) is trivial. It follows that 1f.2(K*) is imbedded isomorphicaily into 'Ji 2 (h) by inclusion, and that a 2cocycle in K* splits globally if and only if it splits locally everywhere. In view oftms isomorphic imbedding we may view elements of ;(l(K*) as idele cocycle classes, and may therefore assign local and global invariants to them. In other words, for c E 'Ji2(K·) we can define naturally inv~ e and inve. We know that the inclusion mapping i commutes with the inflation, restriction, and verlagerung. Hence the properties of Lemmas 1, 2, and 3 are naturally valid for elements of T{2(K*), and the remark made following Lemma 1 is equally valid for elements of 1t2(q, nO). In order to find the invariant of e E T{2(q, nO), it 5Uffices to find its invariant in anyone of the layers in which it splits. In the rest of this section we shall identify 1{2(K*) with the image (under inclusion) in 1t2(JK)' With this identification, the fundamental result (proved below in Theorem 8) may be expressed as follows: Let e E 1{2(JK)' Then inve = 0 if and only if c E 1t2(K*). We begin by examining the invariants more closely in cyclic extensions. Let K/k be cyclic of degree n, with group G. Let G be the character group of G, genera.ted by ;t. The characters take on their values in the rationals mod 1. According to the cyclic theory, any element c E 'Ji 2(JK) is of the form e = x(a)Uc5x where a E JK • If a has pcomponent C1p the local component of c is c" where XP ,..
ReBa, x.
= PrK Resc. x(a) U OX = x{ap) U OX" p
If (ap , Kp/kp) is the local norm residue symbol, then invp x(a) U OX = invl' x(lIp) U c5Xp
== X~«IIp, Kp/k~» = x«ap, Kp/kp», if we interpret (a~, Kp/kp) also as an element of G. To simplify the notation, we shall omit one parenthesis and write for instance x,(ap, Kp/kp) instead of Xp«il~, Kp/kp». We nore that (Ilp, Kp/kp) = 1 at a.ll p where ap is a unit and p is Ulll'8.Il1ified, because a~ is then a, local norm. Define
(a, K/k)
=IT (ap, Kp/kp). p
This product is finite beca.use (a~, Kp/kp) = 1 at almost all p and it is defined because G is abelian 80 that the order of the factors does not matte•.
3. 'RECIPROCITY LAW
We noLe parenthetically that if k = Q is the rational$ and K/Q is cycl0tomic, we have shown in the preceding section that (a. KI k) = O"(a). In particular, (a,K/Q) = 1 in this case. ,We have trivially
(ab, K/k)
= (a, K/k)(b, Klk).
The following identity is obvious:
2)nvpx(Q) UOX = LX(ap,Kp/kp) = p
x(I1(ap,Kp/kp»),
p
p
and oonsequently
inv x(a) U OX = x(a, K/k). THEOREM 6. Let K/k be a normal extension with group G and c E '}i'A(G, K*). Then inv c = o. PROOF. The following argument shows that in number fields, it suffices to prove the theorem in case k = Q is the rationals. Let L:) K be normal over Q. Then ImL c is an element of 1f.2(L/k) and by Lemma 1, invk InfL c = invk c. Acoording to Lemma 3. taking the verlagerung does not change the invariant. The cocycle class Vk / Q InfL c is an element of '/i. 2 (L/Q} and we have
invk c = invQ Vk/Q Inh c.
If the theorem is proved for normal extensions of the rationals, it will follow that invQ V"/QInfL C = 0, and hence invk C == O. 0
We shall now prove the theorem in special cases. Case 1. k = Q is the rationals, and K /Q L'l a cyclic, cyclotomic extension. Any cocyc1e class C E 1{l(K) is of type x(a) U 8X where a E Q. We know that inv ,,(a) U 6x = x(a. K/Q)
IWd we have already remarked that (a,K/Q) = 1. Hence invx(a)U6x = 0, as was to be shown. Case 2. k is a function field and K is a constant field extension (hence cyclic cyclotomic) . K/k is cyclic and unrarnified. Let a E kp • Then lip
= ord, a,
where IfIp is the Frobenius Substitution and (op, Kp/kp) acts on the residue class field locally. If q is the number of elements of ko then the residue class field of p has ql. elements. The effect of on the residue class field is therefore
1fI;'
". ( ql, )'" IfIp:xx
qt.. =x.
and it induces the same effect on the constant field of K. Any cocycle class c E 'H.2(KO) is of type x(a) U Q E k. We look at Q as an ideJ.e, Q = (... , Q, a, a, ••• ), and let vp = ordp a. The produr.l. formuill. gives
ox.
1: /,11, = O. p
VII. RBCIPROCITY LAW
Since
(a,K/k) = ll(a,Kp/kp) ~
we see that the effect of (a,K/k) is identity, namely (a, K/k):
J: +
xqL.!·~·
= x.
Hence x(a, K/k) = 0 and inv(x(a) U ex) = 0, as was to be shown. We now treat the global cases together again, and suppose that k is either 11 function field or the rational numbers. K/k is an arbitrary normal extension. Let C E 'H.2(K). We use K' to denote cyclic cyclotomic extensions of k, and let L be the compositum, L = KK'. In order to prove our theorem, it suffices to prove the existence of an extenlion K' such that ResK ' InfL C = 1. Indeed, K' is a splitting field for InfL c. Using the remark following Lemma 1 we see that Inh c is the inflation to L of a cocycle class r! E 'H.2(K'*)j coupling this with the results of Cases 1 and 2, we see that invII: C o. The existence of K' will be a consequence of the following elementary existence statement.
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PROPOSITION. Glven a finite set 01 primes p, and integers rp (with the obvio'U.8 restriction at Poo that rp"" = 1 or 2 if kp"" is real, and 1 otherwise), there exists Cl cyclic cyclotomic extension K' / k such that IK~ : kp) 's divisible by rp for each prime q I p in K'.
we
Before proving the proposition, indicate how our theorem results from it. invp c will be "# 0 only at a finite number of primes p, and will have denomi~ tors rp a.t these primes. Let K' be a cyclic cyclotomic extension such that (K q' : kp) is divisible by rp for each prime q I p. Actually, [K~ : kpJ is the same for all q I p. Denote this number by mp. Abbreviate InfL C by CL. By Lemma. 2,
inVq ResK' CL By Lemma 1, invp CL
= mp invp CL'
= invp c and hence inVq ResK' CL
=0
for all primes q of K'. This means that ResK' CL splits locally at all primes q of K'. By VI, Section 1, Theorem 2 we conclude that ResK' CL 1. This is precisely what we wanted to achieve.
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PROOF OF PROPOSITION. In function fields we may select for K' a suitably large constant field extension, with highly divisible local degrees at the finite get of primes. In number fields, let t be a prime, and let l" be the l.c.m. of all powers of I dividing the numbers Tp. (We need consider only the finite set of primes t for which i 1.) It suffices to prove that there exists an extension K~ of power degree such that its local degrees at the finite set of primes p is divisible by t". Indeed, the compoeitum K' of the fields K; will be cyclic (because the degrees of its components are relatively prime), and its local degrees will be the product of the local degrees of K;. Hence K' will sa.tisfy the requirements of our Proposition. We let {tn denote a primitive f"th root of unity.
e"
e
a. RECIPROCITY LAW
4.,
e
Suppose is odd. The field Q«(en )(Q is a compositum ohhe field Q(Ct) and a cyclic field K~ of degree enI, II.lld 1Q«(t") has degree ~ (£1) over K~. At a finite prime p this implies that Q,,((in) has degree ~ (e 1) over the completion K~ Qp of K~. By a local result (left to reader) it follows that the degree [K~Qp : Qpj > 00 as n ...... 00. Since this degree is a power of t, we can find an n such that this degree is divisible by tV at a. given finite set of finite primes, as desired. Suppose £ "" 2. Let ( ::;: (z .. with n ~ 3. Let { = (  (1. Let K~ :;: Q({). We contend that K~/Q is cyclic. The automorphisms of Q{()/Q are given by
(1,.: ,
> ,,.
p odd.
Webave (7,.{ :;: (,. 
C".
We note that (2"' "" 1 and one verifies directly that
(1_,.+2"le = U,.{. One of the numbers p. or p. + 2"1 is :: 1 (mod 4). This implies that the automorphisms of Q({)/Q are all induced by automorphlsms (7" where Ii E 1 (mod 4). Trus group is cyclic and the Galois group of Q({){Q is a factor group of it, whence cyclic. Furthermore, (71{ Hence Q({) is not real, and its local degree at a real iofinite prime is 2, as desired. Let p be a finite prime. We know that [iI~M(} : Q"J is a power of 2. Since {Q«() : Q({)] = 2 it followll that I(M() : IQp({)] ~ 2. Since the degree of Q"(O/Q,, increases as n increaaes, it follows that the degree of Qp({)/Qp must also increase. For large n, it can therefore be made divisible by lI.lly given power of 2. Thill concludes the proof of the PropOSition. EJ
= e.
As an immediate consequence of Theorem 6 we obtain a refinement of VI, Section 1, Theorem 2. COROLLARY. Let K/k be normal, and let c E 1{.'l(K·). If Cp = 1 at all primes but one, then c, ::;: 1 at that prime also, and hence c = 1. PROOF. Obvious, because
L: invp c = o.
o
This corollary is a prototype of statements which follow from product (or sum) l'ormul8.'l. We shall meet another one of the same kind later on in the theory. We are now approaching the end of our journey, and a8Ilign invariants to cocycle cla8lle8 of idele cl8.'lse5. We recall the map j: 'H 2(JK) l 'H.'l(GK)' We have defined invariants for elementll c E 'H. 2 (JK ), and the natural way of proceeding would be to assign jc the same invariant as c. However, we encounter the following difficulty: The map j is not always onto. In order to a8Ilign invariants to elements of 'H. 2 (CK) we shall therefore proceed as follows. We begin by making three auxiliary remarks. The first one is essentially a corollary to Theorem 6. REMARK
suppose
1. Let L/k be normal with group GL. Let c E ~(GL,CL)' and d E 'H2(GL' h). Then inVite = inVkd. Proof: We
c = jc = jd with c,
vu.
RECIPROCITY LAW
=
have U(cd l )) 1 and hence c;dl E 1(l(GL. L). By Theorem 6, it follows that inv.\;(cd 1 ) = 0, whence invl;; C :, inv" d. REMARK 2. Let CE i1t 2 (JK), say c inflation commuteB with i, we have
InfL c
=ic. Let L :> K be normal over k. Since
= In£LUc) =j InfLC.
We see therefore that if c is in the image of j in any layer, then its inflation to any bigger layer is also in the image of j. REMARK 3. Let c E 1t2(CK)' Let L1 :J K and L,. :::> K be normal. with groups G 1 and Gz over k. Suppose that
InfLl C = iCl and
InfL. c= i~
=
with Cl E '}i2(G1 , h,) and C2 E '}i2(G2' h.). Let L LIL2 be the compositum. By the transitivity of inflation and by Remark 2 we get InfLC= jlnfLcl =jIn£LC2'
It follows from Remark 1 that invl; InfL C1
= inv" lnfL Ca, and hence by Lemma. 1,
inv/;; Cl = inv" C2
Let K/k be a given normal extension. An element E E 1f(CK) will be called regular if it has the following property: There exists a normal extension L :J K such that InfLE= jc for some cocycle class c E 1{2(GL' h). We denote the subset of regular elements • by"R.2 • We first note that R2 is a group. It suffices to prove that it is closed under multiplication. Let c, d E R? There exists fields L1 and L2 :J K such that InfL. E is in the image of j, and InfLl d is in the image of j. Let L = Ll~' By Remark 2, it follows that InfL C and In£L d is in the image of j, and since lnfL(cd)
=InfL cInfL d
this proves that cd is regular. It proves also that given two regular elements, there exists a common extension L in which their inflation is in the image of j . We shall now define invariants for regular elements. It will be proved at the very end that all elements are regular. Let c E R2 (C K) be regular, and let L :J K be a normal field such that
InfL E = jc with C E 'R2(G L , h). We define inv"c to be invl = 0,
there exists a cocllcle class c E 'Jt2(K) such that invp c = >.... PROOF. Let c E 'H. 2(JK), inve = O. Let c = je be the image of c in '1{2(GK)'
=
Then iov c = 0 and consequently c 1. Hence c is in the kernel of j, which is precisely 1{2(K"). Let now cp E 1f2(K;) be local cocycte c1886e5 having the prescribed invariants Ap. Let c E 'Jt2 (JK ) have local components cp. Then inv c = 0, and by the preceding argument, c E 'Jf.2(KO), thereby completing the proof of our theorem. 0 According to 'Theorem 7 we may now do class field theory in (6, Go) and obtain a homomorphism G +
(a,k)
3. RECIPROCITY LAW
iii
of Ot into a dense subgroup of 0/0'. If a is any ideJe representing a we define
(4, k)
= (a, k).
•
Locally we ha.ve ahlo a class formation and we may form the symbol (Ilp,k,), which is an element of 0p/0~ where 0 p is the local group of fikplk p. We investigate how the global symbol is rela.ted to the local one. THEOREM 9. Let Klk be a finite abelian extension. Let a E C", and let a be any representing idele of a. Then
(a,Klk) =
II, (ap,Kp/kp).
PROOF. The product is finite, because if p is unramified and 4p is a unit, (ap,Kp/kp) = 1. To show the equality it suffices to prove that for every character X of G (the Galois group of Klk) we have
x(a,K/k) = x(Il(a;.,Kp/kp»). p
From the properties of the dual ma.pping we know that
(a,Klk) = inv(x(a) uOX) = ~)nvp(x(a) U OX) p
=
L invp(x(ap) U OXp) p
where
xp = Resc. x.
From the definition of the local dual mapping it follows that
invp(x(ap) U OXp}
Hence inv(x(a) U 6X) =
=x.{Ilp, Kp/kp) = x(ap, Kp/k,).
L: X(Ilp, Kp/kp) = X(n (ap, Kp/k,») p
,
o
as was to be shown.
COROLLARY 1. Let K/k be finite abelian. q an element a e k is a local norm At all primes but one, then it is also a local norm at this prime.
PROOF. This is an immediate consequence of the product formula for the norm residue symbol, 1 = (a,K/k) = Il(a,Kp/kp), II
and of the properties of the local symbol. COROLLARY 2.
o
Let A"lk be the maximal abelian eztension. and let 0 be its
Galois group. Then
(a, k) :::
IT(Ilp, kp). P
PROOF. The product is finite on every finite su bfield of AI ram. Np then N is a neighborhood of 1 in J" and (N, k) c 1). Since Ck has the factor topology, and (k, k) = 1, the corollary is proved. 0
=
=
Let K/k be an a.belian extension. Let Cl e Jk. Then (up, Kp/kp) O'p is an element of the local group Gil' If the idCle It is a. field element 0: E k, then TIp up = TIp(a, Kp/k p) ::: 1. Conversely, given a ret of local automorphisms O'p E Gp, we wish to determine when there exists 0: E k such that (a, Kp/kp) = O'p. It turns out that the obvious nece~ary conditioIlB to be placed on the are also sufficient.
0'"
THEOREM 10. Let K/k be II finite abelian utension 1IIith group G. Gi1le1l a 8et of automorphisms O'p E Gp for each p such that 1. almost all 0'11 = 1 2. I1 p O'p = 1 there exists 0: E k su.ch that (0:, Kp/k~) = a p •
PROOF. By the local class field theory we can obviously find an ideJ.e a E Jk such that (a~, Kp/kp) = O'p. But (c, K/k) ::: fIp O'p == 1 implies that II lies in the kernel of the norm residue symbol, Le. a E k' N J K . We can therefore write a = aNa for some tl E k", II E JK. Locally, (Nap,Kp/k,) = 1 and consequently (tl, K,/kp) == O'p, !!oS W!!oS to be shown. 0
=
The elements a of k' for which (a, Kp/kp) 1 at all primes p a.re k· nNJK, the elements which are local norms everywhere. The conditioDS under which these elements are also global norms will be discussed later in this chapter. 4. Higher Cohomology Groups in Global Fields
Let k be a global field, Klk a normal extension of degree n. We consider the exact sequence
'H. 2 (h) .L.1f,2(CK) !. 'H 3 (K) .!, 'H 3 (h). The third cohomology group in layers of a class formation is trivial. From the local class field theory we know therefore that 'H3(Gp , Kp) is trivial for all primes p. Hence 1f,3(h) is also trivial. This means that
6: 'H2(GK}
+
1£3(K·)
is onto. "}{2(CK) is cyclic of order n, generated by the cocycle class c having invariant lIn. We see therefore that 11 3 (K·) is cyclic. generated by 8c == t, the soca.\led Teichmillier Cocycle. It is easy to determine the order of 'Ji 3 CK"). The kernel, of 0 consists precisely of the images of i, i.e. those idere class cocycles which have an ideIe reprerentative.
4. HIGHER COHOMOLOGY GROUPS IN GLOBAL FIELDS
53
THEOREM 11. Let m be the I.c.m. of all local degrees nR =: [K" : kpJ. The image j'}{2(h) in 'H 2(CK) consists precisely oj those elements E having invariant rim, where r is an 'integer. • PROOF. By selecting swtable components, it is clearly possible to find a cocyc1e class c E '}{2(JK) having invariant 11m. Hence we get all possible invariants rim liS images of idele oocycle cla::;scs. Conversely, it is obvious that these are the only invariants we can get from idele cocycle classes. 0 THEOREM 12. Let Klk be a normal extension 0/ d~ n. Let m = I.c.m. of all local degrees np. Then '}{3(K) is cyclic 01 order n/m, generated by oCKlk where cKlk is the fundamental cocycle class O/'H2(CK)'
We now derive consequences concerning the inflation and restriction of 3rocycles in global fields. THEOREM 13. Let K/k be normal with group G. Let H be a subgroup oiG. Then every element o/'}{l(H. K·) is the restriction 0/ an element of1t.3(G, K*). PROOF.
We have commutativity in the following diagram:
'H. 2 (H,CK) ~ '}{3(R,K*)
Real
lae.
1i2 (G, CK) !1i3 (G, K")
Hence tKIE
o
= OEKIE = ResEOCKlk ~ ResEtKlk'
THEOREM 14. Let K/k be normal, tKlk the Teichmtiller cocycle. There exists a normal extennon L/k, L::l K ::l k, such that lnh tKlk splits. PROOF.
We have commutativity in the following diagram.
1i2(CL) ~ 1i3(L) Iuf
t
f
Inf
1{2(CK) ~ 1t3(K*)
has invariant lin. All we need to do is select a field L having local degrees at least n. Then luiL CK/k has also invariant lin, and is now in the image of j. It is consequently in the kernel of 15, as was to be shown. 0
cKlk
~..
.'
~
.
CHAPTER VIII
The Existence Theorem 1. Existence and Ramification Theorem
=
c::
Let k be a. globallield. Let Cle C, = Co. Let A" be the maximal abelia.n extension of k, and let (5 be its Galois group OYer k. Let w: C _ (5 be the mapping of the norm residue symbol. We want to prove the Existence Theorem: THEOREM 1. Given an open subgroup of finite index B 01 C, there exists an 4belian utension K/k such that B = NCK • K i8 the fixedfie!d ofw(B).
It will be best to prove Theorem 1 in number fields and function fields separately, even though the methods used are quite similar. This will be done in Sections 2 and 3. We shall here simply make further comments on the characterization of abelian extensions of k by their norm groups. First we have an important corollary. COROLLARY. There is a 11 correspondence between the open subgroups of finite, index B of C and the finite abelian extensions Klk, given by B = NCK, (lnd K fixed field of w(B). If Bl and B2 correspond to Kl and K2 respectively, then BJB2 conesponds to Kl n K 2 , and Bl n B2 corresponds to K 1 K 2 • Finally, Kl C K2 if and only if Bl :J B2 •
PROOF. The first part of the corollary is a repetition of the theorem, and of the fact that the norm index is equal to the degree of the extension. The second part is an obvious consequence of these facts. 0
Let k be a global field. One may say that a finite exteJISion Elk 18 a class field if [E : k) = (Crc : NE1,,cE). We have proved that all abelian extensions are class fields, and that all class fields are abelian. An abelian extension is said to be the class field of its norm group. The reciprocity law allows us to determine the ramification of primes in abelian ex:tensions by investigating the idele class group NCK • We imbed each loca.l field kp into Ck via the composition of maps kp ..... Jk . CK. The next theorem gives the connection between the global and local situations. THEOREM
2. Let Klk be an abelian extension,
NCK c,
PROOF.
n k; =
~
,p. Then
N~K~.
In terms of ideles, we have to prove Nh . k· n k; . k· =
The inclusion ::> is trivial. 51>
N1PK~·
k.
VIII. THE EXISTENCE THEOREM
Conversely, let NfJA = 0,/3. Then o:/{3 is a local norm ~t aU primes but p. By the product fannula for the norm residue symbol, we know that oj {3 is a norm at p also. This implies that ap is a local norm, and concludes the proof of the
theorem.
0
The Ramification Theorem now shows how the splitting of a prime p in a class field is reflected in its idele class group. THEOREM 3. Let K / k be an abelian extension belonging to the group B = NCK' A prime p is unrami/ied, in K if and only if Up c B. P splits completely in K if and ani, if k; c B. (If P is archimedean, the two notions coincide, and Up = kp.)
If"
PROOF. is unramified, then all local units are local norms. Hence Up C B. Conversely, if UI> C B, then by the previous theorem Up C (B n kp) NllKll' All local units are therefore norms, and by the local class field theory, p must be Wlramified. The part of the theorem relating to the complete splitting is proved by replacing "Up" by "kp" and "unrami.fied" by "split completely" in the preceding argument.
=
o
The global problem of determining how a prime ramifies can therefore be solved by investigating the norm group, and is completely reduced to a local problem by the preceding theorems. The higher ramification for finite primes will be studied in detail in a subsequent chapter. Note that Theorems 2 and 3 are valid for p finite or no~.
2. Number Fields Let k be a number field. We shall prove the Existence Theorem, and investigate the structure of the open subgroups of finite index in C. We have first: THEOREM
4. The norm residue symbol w: a ..... (a, k) maps Canto !!S.
=
PROOF. We can write C R+ x Co. R+ is infinitely divisible and belongs to the kernel of w. Co is compact and w(Co) = w(C) is everywhere dense in e. It is compact, closed, and hence all of 6. 0
The kernel of w is contained in every open subgroup of finite index. Otherwise some infinitely divisible element would not be in B, and (C : B) would not be finite. Let B be an open subgroup of finite index. Then B is closed in C. Let Bo = B nCo. Bo is closed in Co, hence compact, and (Co: Bo) = (C : B). w(Bo} = ~ is a closed subgroup of 6, and En is the full inverse image of ~ in Co because B contains the kernel of w. Hence (15 : ~) =: (CD: Bo) (C : B). Let K be the fixed field of~. Then NCK is contained in the inverse image 8 of ~ under w. Since (C: NCK) = [K: k] = (15: 1:1) (C: 8) it follows that B NCK. This concludes the proof of the existence theorem in number fields. 0
=
=
=
We now turn to a more detailed study of the open subgroups of finite index.
2. NUMBER FIELDS
Let p be a prime, lip ~ O. If I' is a.rchimedean, ""  0 or 1. Let ap define lip :: 1 (mod p"p) to mean the usual congruence,
ap E Up, up > 0,
e k p • We
if I' finite, lip ~ 1. if I' finite, III' = O. if I' real, lip = 1.
If P is complex, or if p is real but III' = 0, then we put no restriction on lip. By a module m we shall mean a formal product m :: 11p pI', where lip ~ 0, and almost all lip = O. We can think of a module as a divisor in number fields, with archimedean primes entering in it. Let Q be an idele. We define Q == 1 (mod m) to mean ap == 1 (mod 1'''"). Such ideles form a group which we denote by Sm. We let Cm = 6 m k" /k". I
The neighborhoods of 1 in C are obtained from neighborhoods of 1 in J. A fundamental system in J consists of neighborhoods which are constructed as follows:
w= nUp x IIWp pj'nl
plm
where WI' is the group of all up E kp such that u .. == 1 (mod I'"') if I' is finite, and if p is archimedean, then Wp is simply a neighborhood of 1 in kp • The sets Wk constitute therefore a fundamental system in C. The group generated by the elements of W is 6 m if vp = 1 at each real p. Let Np be the group generated by WI" Then Np ;; Wp if P is finite, and Np is either the complex, the reals, or the positive reals if p is archimedean. If 8 is an open subgroup of C then B contains a ne:ighborhood Wk of the type just described. Since B is a group, B.contai.ns the corresponding group Ir m = 1, then 61 = Js where S consists precisely of the archimedean primes. We let k61 = C l . Then C/Cl is isomorphic to the ideal c1wssgI'oup and (C : C1) = h is finite. We may write (0: Cm ) = (0: C1)(01 : Cm ) and (C1 : Cm) is obviously finite because (Up : Np ) is finite for all primes p. We have proved
em·
THEOREM 5. Let k be a number field. Let B be an open subgrov.p 01 C. Then B is of finite index in C, and B contains one of the groups Cm.
Let B be an open subgroup of C. Let fp be the least power of p such that Ill' == 1 (mod f~) implies "l' E B. Then J == TIp fp is obviously a module, and C, c B. In fact it is clear that em c B if and only if f I m. 1 is called the conductor of B. If K is the class field of B, then f is the canductar of K. It is an immediate consequence of Theorem. 2 that fp is the local conductor of K, in the seDSe of the local clwss field theory. We specialize the preceding discussion to the rational nwnbers. A module is given by m = m . p:;" where m is a positive rational number, Poo is the infinite prime, and II ::: 0 or 1. Suppose II I, so m:::;; mp"". Every open subgroup B of C contains a. group Cmp"", The class field to the group Cmp"" is simply the cyclotomic field Q(m) , according to what we have proved in Ch. 7, §2. Recalling the Corollary to Theorem I, this yields Kronecker's Theorem:
=
THEOREM 6. Every finite abelian extension a/ the rational numbef"$ is a cyclotomic field, i.e. is a stJ.bfield aIQ((m) jor some m.
Vlll. THE EXISTENCE THEOREM
Returning to an arbitrary number field, we consider the ~pecial module m = 1. The group C1 is an open subgroup of C and we note that it has been canonically defined. • By the Existence Theorem, C 1 has a class field K/k. By the Ramification Theorem and the definition of CI> K/k is Wlfamified at all primes. (Recall that for an archimedia.n prime, "wrramified" means "split completely".) Conversely, if Kille is an unramified abelian extension, it follows from the ramification theorem that C1 C NK'I"CK" The compositum of unramified extensiOll8 is a1so unramified. We get THEOREM 7. Let k be a number field. Tke m.aximal unramified abeCian extension K of k is finite over k, and [K : Ie] = h is the class number. K is class field to CI, where C1 kJs, S being the set of archimedean primes.
=
K is called the Hilbert Glass Field of k. We abbreviate by RCF. If h = 1, then every abelian extension of k ra.m.ilied, and the HCF is k itself. One may construct a tower of RCF over a number field k. THEOREM 8. Let k he a number field. Let K be its HCF, with group G. Let K1 be the HeF of K. Then Kt/k is normal with group Gl , and K is the maximal abelian subfield 0/ K 1 • In other words, G = Gl/G~ is the factor commutator group
oIG!. PROOF. Let 0' be an isomorphism of Kille. Then (T maps K into itself, so Ktr = K. Since Kf I Ktr = Kf / K is unramified, it follows that Kf C Kl whence Kl is normal over k. 0 Let K' be maximal abelian in Kl' Then K' ::J K. But K' is unramified. Hence K as was to be shown. The Principal Ideal Theorem applies to K and K 1 of the preceding theorem. It was in fact the way it arose historically. . It is an unsolved problem 1 to determine whether there exists towers of HCF with infinitely many steps. More generally it is an unsolved problem to determine whether there exists infinite nonabelian unramified extensions over number fielda. The HCF baa some interesting consequences concerning the class number. K'
c
THEOREM
then
9. Let K/k be the HCF. Let Elk be a finite extension. If EnK
h,,1 hE'
=k
PROOF. K E lEis abelian and unramified, hence is a subfield of the HCF of E. We have hk = [K : k]. By assumption h" = IKE: EJ. It follows that h" I hE. 0 EXAMPLE. Let p be a prime, (1)'' a primitive pI' th root of unity. Then Q( (po) /Q is completely ramified at p. Let F c Q«(p.). Let K be the HCF of F. Then Q(,p.) n K = F. If FeE c Q{(p.), then En K == F and by the preceding theorem, hF \ hE. In particular, the c~ number of the real subfield divides the class number of Q«(p.). We finish our discussion of the RCF by a remark concerning unra.mified extensions in general: Namely, there exists many other unramified extensions besides the ReF. These will of course be nonabelian. One can in fact prove the following result: IThis problem was IIOlved b)' E. S. Golod and 1. R. Shafarevich, see \heir paper
191.
.. PUNCTTON FIELDS
Let we be given any normal extension Klk of a number field with group G. There exist infinitely many finite extensions E / k such that En K = k, i.e. K E / E has also group G, and such that KEIE is unramified.
KE K
/ "E ~/ k
3. Function Fields
Let k be 8. global function field. Let ko be the constant field, with q elements. Let Ko be the algebraic clooure of leo, and let K = Kok be the compositum. Z may be interpreted as the group 60 of Ko/ko, or of Kok/k. Let !p: a  a q be the canonical generator of 0 0 , which then consists of powers !p", v e Z. (We recall that Z is the completion of Z under the ideal topology.) Kok k
/'"Ko
"'/ ko
Let Arc be the ma.ximaJ. abelian extension of k, and let 0 be its Galois group. (J e 0. H (f = !pI! on Ko, we define ord(f = v. We have previously assigned ordinals to idele classes. Namely, if a e C, we can write a = aJ . ao where laol = 1, JaIl = q, v E Z. We let orda = v. Let w; a  (a,k) be the map of the norm residue symbol.
Let
THEOREM 10. Let a E C. Then ord(a, k) = orda. In particular, ord(a, k}
in Z and not in
is
Z.
PROOF. Let a be a representative idele of a. Then from the Reciprocity Law we know that (a, k) TIp(ap, kp) and we view (op, kp) as an element of 6.
=
The global and local ordinals are related by the formula
because the coostant field of k, bas qJ. elements. We have from the local class field theory (.
oM,(O" kp) = OMp IIp.
eo
VUl. THE EXISTENCE THEOREM
It follows that ord(Iip, k,) = 0 at almost all primes, and therefore ord(a, k} = ~ ord(ap, k,) 9
= L!pordp(Clp,kp) p
=
L /, ord
p IIp
p
=orda
o
thereby proving the theorem.
COROLLARY. Let ~o be the group 0/ Ak/Kok. Thenw is a topological isomorphism of Co onto 1)0. 0 is isomorphic to the direct product of ~o and 0 o . PROOF. w(Co) is contained in 1)0. Conversely, if w(a) E ~o then orda = 0 and a E Co. Hence w is an isomorphism of Co onto 1)0. It is continuous, and since Co is
compact it is a homeomorphism onto. Let 11' E 0 be a lifting of a topological generator r.p of 00. Then the closure (0) of the subgroup generated by 0 maps isomorphicalJy to ~o, and i5 = (0) X 0o, 0
Let ordal = 1. Write C as a direct product {all x Co. Any a E C can be written a = al x ac. We see that C is isomorphic to Z x Co (but not canonically). We define a new topology on C: The neighborhoods of 1 are to be the open subgroups of tillite index in the ordinary topology. This topology will be called the
class topol09Y.
•
LEMMA. The class topology coincides with the old one on Co. It inducc:.B the ideal topology on Z and the product topology on Z x Co. PROOF. Let B be an open subgroup of finite index in C. Let b E B have least positive ordinal. Let Bo = B n Co. Then B = UI'EZ b~ Bo. Eo is open because Co is open. We shall now discuss those open subgroups, and prove they are of finite index in Co. As in our discussion of number fields, we let m ;: p p", (/lp ;;;.. 0 and =: 0 for almost all primes) be a module. A fwldamentai system of neighborhoods of 1 in J is given by the groups
n
6 111 =
nUll pfnl
x
I1N"
plm
where Np are the groups of 01' E kp such that 0" == 1 (mod p"'), for P I m. The groups C m = k6 .. are a. fundamental system in C, and in fact C m C Co. We have
(Co: em) = (Co: Uk)(Uk :'C",),
nail
=
where U = P Up. By the finiteness of class number, (Co: Uk") h is finite, and obviously (Uk: Cm ) is finite, because (Up: N p ) is finite. The open subgroup Bo must contain some open group em and hence (Co: Bo) is finite. It is clear that the induced topology on Z is the ideal topology, and this proves our lemma. In addition, it gives insight into the :;tructure of open subgroups similar to that obta.ined in number fields. 0
'I
3. FUNCTION FIELDS
af
We form the group 0 = x Co ~ Z x Co by taking the formal Cartesian product {an x Co where the exponents 1/ of a] now range over Z. Both Z and Co are compact, so C is compact. If we look at the effect of the nonn residue map on C, we see that w is uniformly continuoUll, in the new topology. We may therefore extend w by continuity to C. Since C is compact and w(O) is dense in ~, it follows that w(O) is onto 16. w is an isomorphism between C and 16, and consequently w is a homeomorphism. We can now prove the existence theorem in function fields using the same procedure that was used in number fields. Let 8 be an open subgroup of finite index inC. Bo BnCo, B Ul'Ezb"Bo. If we write b = af then (Z: dZ) = d. Let 73 = dZ x Bo. Then
=
(C : B)
= d(Co : Bo) =
=
(0: 73).
Let ~ = weB). ~ is closed in 16 aDd (C: B) = (16 : ~). Let K be the fixed field of Then [K: kJ = (C: B). We know that WeNCK) C~, and hence NCK C B. Since (C : NCK) = (K : k] we conclude that NCK = B, thereby proving the existence theorem. As in the local class field theory, we remark that the open subgroups of finite index in C are in 11 correspondence with those of C. Namely, let 8' be an open subgroup of finite index in O. Let Bo := 8' n Co, and let r B' n Z. Then I' is a closed subgroup of finite index of Z, and is consequently of type dZ where d is an ordinary integer. We see therefore that 8' is simply the closure of the group d:l. x B o. The Hilbert C1&!s Field has a certain analogue in function fields. The maximal unramified abelian extension of k is infinite because of the possible constant field extensions. We may ask however for unramified abelian extensions which have the sarne constant field. Let Cl == Uk·/k·. Let b E C t ordb = 1. Then the group B = {b} X Cl is open in C, and (C: 8) = h is the class number. The class field of B may be viewed as a HCF. It is a maximal unramilied abelian extension having the same constant field as k. We contend that there are h such class fields, and that their Galois groups are isomorphic to C/B. Let UI, ..• , Uh be representatives of Co/C1 • Let b; "" Qui. The groups 8j = {bi} X C 1 are all dL~tinct and (C : B i ) = h. Furthermore the factor groups C/Bi are all isomorphic. The groups Bi give rise to h class fields of the above mentioned type, of degree hover k, with isomorphic Galois groups. Let orda = 1. Then ordab 1 "" 0 and hence a is in some coset ~ mod Cl' This proves that the HCF we have described account for all of them. We denote these HCF by K j , i = 1, . .. ,h. The class field of {b k } X Co is the constant field extension of degree h. Denote it by L. We contend that LK; = LKj = LK; ... K" for any it j. Indeed,
weB).
=
{iI'} x Co n fbi} x C1
/I.'
11
= {bf} XCI,
and are in the same coset mod C1 . This proves our contention. The field K = LK1 •.. Kh is of degree h2 over k. It is invariantly defined and is may be viewed as the correct generalization of the Hilbert Class Field. It is class field to {filL} X Cll where b is any ideIe class of ordinal 1. On the other hand, noting that in number fields the HCF is class field to the group k* Js"". where Soo is the set of a.rchiroedea.n primes, one could say that an
But
VIII. THE EXISTENCE THElOREM
analogue in function fields is the class field to k' Js, where S].s any nonempty set of primes which we designate as the "infinite" ones. Then the Galois group of the Dedekind ring as of functions with no poles outside S, in strict analogy with the number field case, where the Galois group of the HCF is isomorphic to the ideal class group of the ring of integers. The constant field extension in this type of HCF is of degree equal the qcd of the degrees of the primes in S.
"\
4. Decomposltion Laws and Arithmetic Progressions
We coIlllider only number fields k, and leave to the reader the task of formulating
the analogous results in function fields. Let In = n~ p'" be a module in the sense of 2, and G p = k·S In • Let K be the class field of G m• Let Jm be the group of ideles having component 1 at all p I m and arbitrary otherwise. We contend that every idele class mod k"6 m has a representative in Jm • Indeed, let a be an idcle. By the approximation theorem, there exists 0: E k such that aa == 1 (mod p"p) for p I m, vp ~ 1. This implies that CIa E J",6 m , or in other words, J =' k4 Jm6 m as contended. By an elementary isomorphism theorem, we have
J/ke", ~ Jm/Jmn k*6,..
Let tim: J map ideJ.es onto ideals by putting t",(a)
+
J
= ii = IT
pord._. We get a map
pfm
pftnite
tm:
JmJ",
of J m onto the ideals prime to m. Let k~ be the principal ideals represented· by elements Q E k*, 0: == 1 (mod m). Then ofJ;l(k~) = Jm. n k*6 .... Hence we get
Jlk"e m~ J... /Jm nIt."6m::= Jm/k:". Let G be the group of K/k. Let w: Jm+G be the norm residue map restricted to Jm • Then w depends only on cosets of J mn k*S",. For every ideal ii in J", we select an idcle 11 E J m such that ofJm(n) = ii, and define (if) to be weal. We know that w depends only on oosets of J m nk*6., and consequently
(f)
depends only on c05ets of k~. The map
is a homomorphism of Jm onto G, with kernel km . We interpret ilk:' as a generalized arithmetic progression modm. Let p be a finite prime. It follows from the Ramification Theorem that if p f m then p is unramified. Let '/I" E kp , ordp 11" = 1. The idCle 'lI" = ( ... ,1,11",1, ••. ) having 'If at p and 1 at all other primes is such that 011",(71") == p, and hence w('/I") = (f) is the Frobenius Substitution. This shows how the norm residue symbol can be described just in terms of the unramified primes. If q is another prime we have =0 (~) if and only if p and q lie in the same arithmetic progression.
(f)
of. DECOMPOSITION LAWS AND ARrl'RMETIC PROGRESSIONS
, e3
If k = Q is the rationa.ls, then a module m is of type m or mpcx> where m Is a positive rational. Considering the latter type, m = mpoo. we see that 4>m maps J mpao onto the ideals prime to m mod m. A rational number Q is == 1 (mod m) if and only if it is positive and :: 1 (mod m) in the ordinary sense. This shows that two integral ideals are in the same generalized progression mod mpoo if and only if their unique positive generators are in the same arithmetic progression mod m. Thus our "generalized progressions" do generalize the cl/lBSical ones, which justifies our terminology.
!
.~
CHAPTER IX
Connected Component of Idele Classes 1. Structure of the Connected Component Let k be a global field. Let 6 be the Galois group of its maximal abelian extension, and let w: Ck ..... 6 be the map given by the norm residue symbol. In function fields we have seen that the kernel of w is trivial, Le. consists of 1 alone. lu number fields. the kernel is the subgroup of q, of all elements which are infinitely divisible according to Prop. 10 of Ch. 14. We shall prove that this subgroup is the connected component of identity of Ck, which we shall denote by Dk' We shall also determine the structure of Die by finding a set of representative ideles whose images generate Dk. We need an auxiliary result (which will reappear in sharper form in the next chapter). We prove a little more than is immediately needed for our purpose. We denote by a primitive mth root of unity.
em
THEOREM 1. Let k be a global field, m = 2tm' (m' odd) an integer and S 1& finite set of primes. Let Q E k and assume a E k;m for all p ¢. S. A. If k is a junction field, or if k is a number field and the field k«(2.)/k is cyclic (this condition is certainly satisfir;4 if t ~ 2) then Q E k*m.
B. Otherwise at leIJst Q
E
k· m / 2 •
PROOF.!. Suppose m and n are relatively prime, Let rm + 8n = 1 then
P. 1 ill k.
Q
= fJ'" 8Ild a =
"I" with
This shows that it suffices to prove our theorem in case m pr a power of a prime and we shall assume this for the rest of the proof. 2. Let p be the characteristic of k. Consider the inseparable extension K = k(y'a). Since f(ffl C k, any valuation p of k extends uniquely to K which means that only one prime ~ of K divides p. From the elementary theory of global fields we know L.  c), (for a fixed E) is a continuous map of V into U. If PI, ... ,Pro are the complex primes of k, we have unit circles in each of the corresponding local fields. We describe the ~th unit circle by the ide!e tPl'(t,..) which has at p,.. the component e 2".it" (tl' E R) and 1 at all other components. Let Cb C2, ... '''r be a system ofindependent totally positive Wlits of k (if ei is not totally positive its square will be).
•
IX. CONNECTED COMPONENT OF IDELE CLASSES
We note first that the ideles of the form
(2)
c~lci· ... c~r¢l(tl)" .tPr.(t... )
Al
E
V, t, E R
form a group and have V()iume 1. We want to know when is principal.
all
idele of the form (2)
LEMMA 1. Let ~ = (Xi,Si)' An itUle of the form (2)is equal to an element a of k if and only if each Ai and each ti is an integer. For the A. this means that Xi is an integer (i.n the natural imbedding in Z) and that Si is the same integer.
PROOF. The sufficiency of the condition is obvious. Suppose conversely that the idele of (2) is = Q E k. Looking only at the finite primes we must have
tf' ... c;r = a.
(3)
The element a must be a unit. ad. lies therefore in the SUbgrollP generated by the Ei, say ad. = E~r with I'i E Z. Raising (3) into the dth power we obtain:
et' ...
ef'I'l ... e'/.X r I'r = 1. Theorem 2 shows that dxi lSi = O. This means (if one considelll the ordinals of Xi at the primes p of Z) that the integer J.Ii is divisible by the integer d in the ordinary sense, and since d is not a divisor of 0 in Z, this means that Xi is an ordinary integer. This makes now I! =
Ei' ... e;r
is equal to a a.t all finite primes. Even if we had this only at one prime we could conclude a = c. The infinite components of a are therefore ~'e~2 ... ~r on one hand, whereas (2) gives t1't~· .. .t:r¢l(tl).' .¢... (t r.). The relation is:
an element of k. This
(4)
E
Ei,Z1erZ, ... e:,.a:"4>I(tl)."¢",(4.) = 1. If we take components at Poo and then the absolute value we obtain lell;~""
... IErl~::"'r = l.
X,.
From the independence of our units we see immediately that IIi = If we substitute this result in (4) it follows now trivially that all the ti are also integers. This 0 proves the lemma.. The reader is invited to investiga.te how it comes that the roots of unity of k did not play a role in our proof. Consider the map
P'l .... ,Ar,t!, .... t ... ) t etl ... e:
r
cfJl(tl) .. . cfJ... (t,..).
We shall use the vector notation, and abbreviate this map by
(A, t) ...... E),rjJ(t). It is a homomorphism of V x af'2 into the ideIes and it is continuous. We may follow this map by the canonical homomorphism into the idele classes. According to the preceding lemma, the kernel consists precisely of zr x zr.. Each Z is closed in V or R, and we obtain a continuous isomorphism of (VIZ)" x (R/Z)'" into Co. It will be shown below that it is a topological isomorphism onto the connected component of Co. The following lemma concerning the topology and the group structure of V /Z will be useful. r
1. STRUCTURE OF THE CONt4ECTED COMPONENT
LEMMA 2. VIZ is compact. The. reals JR, naturally imbedded in VIZ, are everywhere dense in it, and VIZ is consequently connected. VIZ is infinitely and uniquely divi.sible.,.. , PROOF. 1. One sees
easily that every element of VIZ Iw; a representative
). = (z, s) with 0";; s ,,;; 1. If we denote by N the set of all" = (0, s) with
°
~ 8 ~ 1 then our representative is in Z + N and this means that VIZ is the image of the compact set Z + N under the canonical map of V onto VIZ. V jZ is therefore compact. 2. Let). = (:c,s) be any element of V. If m is a given integer (describing a neighborhood of Z), let I' == x (mod m), I' E I. and Bubtract the image of IJ. in V, so the pair (/J, /J) from A. This leads to the pair (x  /J, S  IL) which repre.o;ents the same element as Ain VIZ. This pair belongs to the set (mI,O) + (0, 8  J.t) which is mapped into a neighborhood of the real s  /J in VIZ. Therefore Ii is dense in V /'1... Since the closure of a connected set is connected, it follows that VIZ is connected. 3. The same computation allows us to write" == m(~,~) (mod Z) and shows divisibility by m. If mA == 0 (mod Z) then there would be an integer IJ E Z such that mx = IJ. in Z and ms = IJ. in JR.. The first equation implies (as in the previous lemma) that x is an integer and the second shows now that s is the same integer. But this means" == 0 (mod Z), and proves the unique divisibility. 0
The group VIZ is called the solenoid. It can be shown that it is simply the compact dual of the discrete additive group Q of rationals. We return to our continuous isomorphism of (VIZ)" )( OR/z)ra into Co. Each circle R./Z is compact and infinitely (but not uniquely) divisible. It is obviously connected. The group (VIZ)" x (JR./Z)'"" is therefore compact, connected and infinitely divisible. Its image Do in Co is therefore compact, connected and infinitely divisible and our map an isomorphism, both algebraically and topologically. We contend now that every infinitely divisible idele class of volume 1 lies in . Do. Let a be such a c1a,.;
III I;;. . , .. ·1E'rl~+I .
=(0, 5i) e V. Since ii is totally positive, we have iip""
= (~' ... f~~)p""
for any real 1'"". At a complex prime they differ by an element of value 1. Hence we can write
IX. CONNECTED COMPONENT OF IDtLE CLASSES
thereby proving our contention, that Do contains the infinitely divisible idele classes of volume 1. Since Do itself is infinitely divisible, it is therefore equal to the infinitely divisible idele classes of volume 1. This shows that Do is the intersection of all open subgroups of finite index in Co. It contains therefore the connected component of identity in Co. Since Do is connected, it now follows that it is tbe connected component of 1 in Co. In order to get the full connected component of C, we select at some archimedean prime 1'00 a positive real line IR+ C kp~. Then it is clear that every element from the connected component is uniquely represented by an idele of the group
lR+ . f:>' • ¢(t). Thus we have proved THEOREM 3. The mapping (A, t) + e'q,(t) of (Vlzt x (R/Z) .... into Co is a topological isomorphism onto the connected component of Co. The structure of the connected component of C is that of a direct product of one rea/line IR.+, r :; rl + r2  1 solenoids, and r2 circles and is the clo~re in C of the image of the connected component of J.
2. Cohomology of the Connected Component Let Klk be normal with group G. As we shall deal only with the ideJ.e classes of K, we let
C=CK,
D==DK,
Do=DKnct.
Then, as is ea.sily seen, it is possible to choose a splitting such that D = Do x lllt and G has trivial action on lR+ . Let {13} range over the complex primes of K. The circles ofthe preceding section are now denoted by cPll(tll)' Such a circle is the idele having all components 1 ~t the 13component, which is e2"it~. We can write
IT ¢1l = rr(rr ~).
'.P P !llip Indeed, each qJ of K extends exactly one prime p of k, which may be real or complex. It is clear from the definition of the action of G on ideles that each semilocal component II lllP ~ is invariant under G, and hence the full product is invariant under G. (This comes from the fact that each local module ~ consists precisely of those elements of the local complex field having absolute value equal to 1.) Let D' be the subgroup of D given by the canonical image of the circles in C, i.e.
D' =
IT ~(tll)K.
'.P Then D' is Gisomorphic to the product II
rnD')
+
'W(D) 1r'(D/D')
we get THEOREM
4. For all r, 'W(D')
~
'W(D).
n
2. COHOMOWGY OF THE CONNECTED COMPONENT
The cohomology of lY is now ellSily determined. For each semilocal pair (G, n~lp ,) be a local component. From the scmiloca.I theory, we know that • 1i" ( G, ~ 1i"(Gp, 4>,),
ntkJ)
~Ip
and since the cohomology group of a product is the product of the cohomology groups, we have 'W(D') ~ W(G" 41,,).
n p
If p is unramified, i.e. splits, then Gp is trivial, and so is the local cohomology group. If P is real, Gp is cyclic of order 2, generated by the complex conjugation. From the cyclic theory, we know that the cohomology groups are periodic of period 2, and hence our computation is reduced to the dimensions 1 and O. In dimension 1, we know that the cohomology group is isomorphic to the elements with nann 1, modulo those which have obviously nann 1 (i.e. t.hose of type a1  u ). One verifies immediately that for the circles, this factor group is trivial.. In dimension 0, we know that
'}{J(G p , rI>;.} ~ (tP~, : NptPp), i.e. the fixed elements modulo the norms. But the fixed elements are simply ±l, and the norms are trivial, l.e. equal to I. Hence the factor group is cyclic of order 2. Thus we have proved: THEOREM 5. Let K/k be normal with group G. Then 1{2r(D') is of type (2,2, ... , 2) and order 2'" where IJ is· the number of ramified archimedean primes. '}t2r+l(LY) = I lor all r.
In the exact sequence
1 = 7£l(C) ~1{l(C/D) ~ ~(D) _'H2(C) +7£2(C/D)
+
1
the map i maps each local factor of 'H2(D) onto the kernel of multiplication by 2 in the cyclic group 71 2 (C). Thus, if J.l. > 0, then 71 1 (C/ D) is of type (2,2, ... ,2) and order 2"1, and CH2(C/D) is cyclic of order ![K : k]. We note that f{l(C/D) is not necessarily trivial, and that consequently the inclusion i: 1{2(D} + 1{2(C) is not always an isomorphism.
CHAPTER X
The GrunwaldWang Theorem l 1. Interconnection between Local and Global mth Powers
Let k be a globaJ field, m any integer and S a finite (possibly empty) set of primes. Denote by P(m, S) the group of those elements a E k' which are in k;m for all p ~ S. kom is a subgroup of P(m, S) and we ask now for the precise structure of the factor group. Theorem 1 of the preceding chapter provides already an answer in certain cases and we must now investigate the remaining possibilities. If k is a function field, then P(m, S) = kom • Assume therefore from now on that k is a numberfield and let m ::; 2cm' (m' odd). The answer is again P(m, S) = k m if k«2' )/k is cyclic (5ee IX, Section 1, Theorem 1). We are therefore led to investigate the conditions under which k«2C) is a noncyclic field. Changing the notation slightly d~ote by ~.. (r = 1,2, ... ) a primitive 2~th root of unity such that (1)
~+1 ={,..
Put: (2) and observe that 'fir
(3) (4)
= 0 if and only if r = 2. We also obtain: r)~1 = 2+1'Jr ( ..+1I'/r+1 = CO' + 1.
(3) shows that a field containing 1/.. also contains all "1/l with p. ~ r. (4) shows that a field containing 7]r+l and (r will contain ( ..+1 if r ;;;. 2. If therefore (2 = i and 7Jr{r > 2) are in a field, then (r is also in this field. k« .. ) is therefore the compositwn of the fields k(i) and k(1Jr) for r > 2. lin 1932 W. Gru.nwald proved a theorem like Theorem 5 of this chapter in hill dissertation (supervised by H. Haase, d. 110)). However he did not notice the need for an extra condition in a certain special case. Ten years later, G. Whaplell published a new proof of Grunwald's theorem {30! al50 overlooking the special case. This new proof was presented by Bill Mills in Artin's seminar in the spring Df 1948. A few days later, one of the listeners, Shianghaw Wang, came to Artin's office with a counterexample to a key lemma of that proof, and later the same day produced a counterexample to the Theorem itself, by proving that there does not exist a cyclic extewion of Q of degree 8 in wbich 2 stays prime (see the 'consequence' before Theorem 2 below). In his Ph.D. thesis, 126J Wang published a corrected version of Grunwald's statement, which is now known 118 tbe GrunwaldWang Theorem. This chapter is Artin's own reworking of this tbeorem, ita proof and related questions, in the light of Wang's discovery. 13
X. THE GRUNWALDWANG THEOREM
The extensions k(1/r)/k are cyclic. To see this it is sufficient to show that QC1/r)/Q is cyclic since the group of k(1/r)/k will be a subgroup. Every a.utomorphism ofQ«(,.) is induced by an automorphism a"C'r) = Cf of Q(TJr) and (2) shows that ajJ and a_". will induce the same automorphism of Q(1/,.). We may therefore
Q«. }
fOl'm a cyclic group assume that ~ == 1 (mod 4). These automorphisms 17". of generated by Us and this shows that Q(J).. ) is cyclic. If k would contain all TJr then k(i) would contain all which is impossible. There is therefore an integer s ~ 2 such that 1/. E k but 1/.+1 r¢ k. Because of (3) k(1/.+1)/k is quadratic. Suppose now that all k{(r)/k are cyclic. k(f/a+1) is then the only quadratic subfield they can have. k(i) is either k 01' quadratic and consequently i E k(1/.+1). If conversely i E k(J).+ll C k(flr) (for r;. 8+ 1) then k«,.) == k(TJr) and is therefore cyclic. The noncyclic case is therefore characterized by the fact that F = k(i, fI.+d == k«(.+l) is a four group field over k. In this case, F contains the three quadratic subfields k(i), k(718+1) and k(i1/.+l) and (3) shows that this fact can be expressed within k by the condition that the three elements 1,2 + fl., (2 + fl,,) are nonsquares in k. Since fl. E k, k(i) = k«(.) and we see that k«(e) is cyclic for t ~ s. But F = k('Hl) and all k«(d with t > s are noncyclic. Let a be the automorphism =F 1 of '(i)/k. (;1 fI, E k and (.(;1 = 1 E k show that (. and (;1 are conjugate: (: = (; 1. Suppose now that there exists an Q E P(m,S) which Is not in k m . Then Q f/. P(2t, S) but Q E ,._2' (since Q E kom'). We see immediately that we must have the noncyclic case and that t > 8. • With k(i) 118 ground field we have the cyclic case so we can write Q "" AZ'. Raising to the power 1  a we obtain:
'I"
,,+
=
(A I a )2' == 1.
Ala is a 2tth root of unity in k(i) = k(,,) and (,+1 f. k(i) (it generates F). Therefore Al en.
=
I. ABELIAN FIELDS WITH GIVEN LOCAL BEHAVIOR
7!1
PROOF. 3 Both groups PoG" and PC" are closed in G. Since Po is of finite index in P, PoC" is of finite index in pen. The complement of PoC" in PC" consists of 'a finite number of (closed) cosets of PaC" and is therefore closed in the topology of PC", PoC" is therefore open in the group PC". This means that there exists a neighborhood V of 1 in C such that pen n V c PoC". Put N = PaC"V. According to Lemma 6, N is an open subgroup of finite index of C. PO"nPoO"V = poen(PC"nV) == PoC". Therefore PnN = PnpC"nPoCnv =
PnPoC" :;:po(pnCn ).
c
I /"" N /"" / PoC" PN
PC"
P
...
"
o
= PoC"V
"'/""
Po(p n C") ::: P n PoC"
PO" n V
THEOREM 4. Let Po be an open sv.bgroup of finite indez of P. There exists an open subgroup N of finite indez of C such that P n N = Po. The smallest integer n that one can achieve as exponent of the factor group GIN is the smallest n for which P n 0" CPo. 1Hvially PNIN ~ PIPo. Let m be the exponent of PIPe. Then n = m if P n cm c Po; otherwise, n = 2m. PROOF. 1. Suppose an N is found such that P n N == Po and such that the factor group GIN is of exponent n. Then C" C N whence P n 0" c P n N = Po. 2. Suppose P n 0" c Po. Using Lemma 7 with this n we find an N such that
PnN=po. 3. Since Po is of finite index there is an n such that P" circ:umstances P n cPn C pn C Po.
C
Po. Under any 0
This result suggests that one flhould impose further global conditions. We study first the special case when PIPo is a cyclic group of order m. We may view in this case Po as the kernel of a continuous character Xs of P of period m. The restrictions of xs to (if PES) define local characters XP on whose kernels &re Po n k" (i.e. the local nonn groups of the global field K belonging to N). If 7tp are the periods of the Xp, then m i.s the least common multiple of the "p' If the X" are given then Xs llpES Xp· The question arises whether XS can be extended to a continuous character X of C and what the minimal period of such aD extension would be. The kernel N will be an open subgroup of finite index of C and the exponent of the factor group will be the period n of x. Theorem 4 shows that the minimal value for n that one can hope to achieve is n m unless we have P n rt. Po when n = 2m will be needed. Let n be defined in this way.
k;
k;
=
=
em
3We make use of the following lemma: If B is .. group, A a sublet of B and C any o&her set then BnAC= A· (BnC).
/
80
X. THE GRUNWALDWANG THEOREM \
According to Theorem 4 we can find a.n open subgroup Nl of finite index of C such that Pn Ni = po. PN1/NI ~ PIPo and such that CINI has exponent n. We conclude first that XS can be extended naturally to a character of the group P NJ with kernel NI, If we view this extension as a character of tbe finite group p NIlN 1 , we can extend it to a character of the finite group C / N l • If we view this extension as a character on C it will be an extension of XS to C, will be continuous since its kernel N contains Nl and will satisfy xn = 1 since C/Nt has exponent n. Finally we look at the condition that determines the value of n. Lemma 2 shows (since pm C Po) that we have n = 2m if and only if we are in the special case for P( m, 8) with a nonempty set 8 0 and if the ideJe class em of Lemma 2 is DOt in Po. In other words if Xs(c m ) :; npEso X,,(O'o) == 1. We have proved:
THEOREM 5. Let 8 be a finite set of primes, XP local characters of periods nl> lor each PES and m the least common multiple of the np. There exists a global character X on C whose local restrictions are the given XP' lts period can be made m provided that in the special case the condition:
=
IT X.(ao) = 1 IIESe
is lJatisjied. If the condition is not satisfied one can only achieve the period 2m.
We describe very briefly the corresponding question if PI Po is abelian. Let Xl, X2, " . ,Xr be a basis for the group of characters of P/ Po. Let ei be the period
of Xi. The question is whether one can find an N such that C / N :::! PI Po and P n N = Po. This turns out to be equivalent to the condition that each of the Xi can be extended to a character x. tif C with period e.. If this is possible, one takes for N the common kernel ofthe Xi. We return now to the cyclic case.
REMARK. Assume that we are in tbe special case, p € So and K,,/kp the local extension described by the given character XI> of period np. If TIp is even, let K~ I kp be its quadratic subfield. Then Xp(eto) = ±1 where the  sign occurs if and only if ::; is odd and (2 + 7]1J) is not a norm from K;/kp, PROOF. According to (5) eto = ((2 + 1/8»"'/2 , and consequently X,(O:o) = 1. If ::. is odd we write:
X,,{o:o) =
If;; is even, then 0'0 E k;'
(xl ((2 + 71.») ~ .
X;e. is the character of period 2 that describes the field K~ / k., so that the exponent ::;. This proves the remark.
we can drop 0
LEMMA 8. Assume that we are in the special case, that n ill even and pESo. There exists a cyclic local extension of degree n" such that (2 + '18) is a norm from its quadratic S'Ubjield,
PROOF. It suffices to achieve 11.+1 E Kp. Indeed, K; would be = kp(1/.+I) and NK.lk.(f/s+l) = ~+1 ::: (2 + 1/,). To find such a K II , observe that the fields k p (ll,,), I.t. > s are all cyclic and have as degree a power of 2, Among them there are fields of arbitrarily high degree since a given field will contain only a finite number of the Ti". All of them contain 1/8+1' It suffices to take as K the compositum of 0 such a field with an unramified field of suitable odd degree,
I. CYCLlC EXTENSIONS
al
We can now prove a theorem that has applicatioWl in the theory of algebras over global fields: THEOREM 6. Let k be a global field, 8 a finite set of primes and flp positive integers associated with each p E 8. If P is archimedean np should be a possible degree for an extension of k~. Then there exults a cyclic extension K I k whose degree " is the least common multiple of the "p, and such that the completions K'Jl/kp have degree flp for all pES. PROOF. To each PES select a cyclic field Kp/kp in the following way: 1. If P is archimedean select the extension in the only way that is anyhow possible. 2. If P is nonarchimedean but not in 80 select the unramified extension of degree np. For primes in So select the extension as in the previous lemma. H we describe each of these local extensions by characters, we can find a global character of period n whose restrictions to kp are the given local ones since we have taken care that in the special case Xp(Qo) = 1. 0
The following corollary is directly adapted to the intended application to algebras: CoROLLARY. Let k be a global field and c E 1t2(1!}, 11) a 2cocycle of period n. Then c has a cyclic splitting field of degree n over k. PROOF. We know that c p = 1 at almost a.ll primes. The period of c is the least oommon multiple of the periods of the cpo Each Cp is split by any local field of degree nIl' Furthermore c splits globally. if and only if it splitsloca.lly everywhere. By Theorem 5 we can find a global cyclic field K of degree n which has local degree Rp at all primes where Cp ~ 1. This field will split all Cp and consequently c. 0
n,
3. Cyclic Extensions
It is possible to ask a somewhat different question. Namely, given a cyclic extension Klk of degree n, and a prime p. We wish to determine when there exists a field L:> K which is cyclic of degree y over K, and alw cyclic over k. If p then the question reduces to the one already treated. We suppose that pIn. To Simplify the notation, let C denote the idele classes modulo the COWlected oomponent in the case of number fields, and the compa.ctified idele classes in the case of function fields. We have therefore C "" C in the notation used previously. Denote by 8 the character group of C. Let N be the norm group of K, to which K is class field, and Np = Nnk,. Let X be a character with kernel N. A field L of the prescribed type will exist if and only if there exists a character", E 8 such that X "" 'fj;Pr, or in other words, X E {jPr. Indeed, the kernel of such a character '" would be an open subgroup MeN of finite index prn in C, and its class field L would satisfy our requirements. The subgroup of C orthogonal to is precisely 11 / 1'r (the elements of order p dividing pr in C) because a E (C ")· # tP'r (a) = 1 for a.ll '" E 8 # ¢(apr ) = 1 for all '" E 8 apr = 1. In function fields we note that the elements 11/1'" have ordinal 0 and hence are senuine idele classes.
tn,
tv
X. THE OlWN'WALDWANO'THEOREM
The following theorem is essentially trivial if we take into occount Theorem 2 and the relation between the local and global groups Np and. N described at the beginning of Section 2. THEOREM 7. Let the notation be that described in the preceding paragraphs. The Jollowing statements are equivalent: 1. There exists an e3:tension L oj K soch tOOt LI K is cyclic of degree pr and Lll is also cyclic.
2. X E
Cpr.
3. At each prime P, X,(C> = 1 for every pT th root oj unity ( in l p , and in the tpecial C48e, );(c"r) = 1 (where Cpr is the icUle class oj Lemma 2). 4. At each prime p, , E Np Jor every pr th root oj unity in k" and in the apecial ca8e, Cpr E N p • COROLLARY 1. If 1 is II junctioo field and p is the characteristic then the construction oj Theorem 7 is alway~ PQssible.
PROOF. There are no idele claBses of period p. and there is no special case.
0
COROLLARY 2. IJ k is a global field and ( is a primitive p" th root of unity in k, then the Jour statements oj the theorem are equivalent to the condition that ( is II nQrmjrom K, i.e. (= No., c:r E K. PROOF. Since' lies in k, there is no special case.
H,
is a global norm it is
Ii
local norm everywhere. Conversely, if , is a local norm everywhere, it is a global norm because Klk is cyclic. This proves the equivalence of our statement with 4 and hence with the others. 0 The conditions formulated in Corollary 2 and in Theorem 7,1, are completely algebraic, i.e. do not involve arithmetic objects. It can be ShOWIl that they are equivalent in any abstract field k. Corollary 1 is likewise true over any field 'k of characteristic p. See for instance N. Bourbaki, Elements, Ch. V, Algebre, 11, Exercises. We see that the algebraic theorems and criteria are special cases of the arithmetic theorems for global fields, but that the latter hold whether the roots of unity are in 1 or not.
CHAPTER XI
Higher Ramification Theory 1. Higher Ramification Groups We recall some elementary facts of local valuation theory, and begin by fixing the notation. Let k be complete under 0. discrete valuation, with perfect residue claBs field k. The characteristic of k is p ~ O. Klk denotes a normal extension The prime ideals are \lJ and p respectively, the rings of integers 0 and o. IT has order 1 in K, and 71' has order 1 in k. We write Q N {1 if a and {1 have the same value. Elk denotes a finite separable extension. "EI" is the different. Elk is 11lll'amified if and only if f}Elk = 1. If K :J E :J k then 'lJK1k = "X/E' f}E/k' If the powers of a are a minimal basis for Elk, and f(x) is the equation for a over k, then iJ E / k t{a)O.1 In a tower K :J E ::> k, we distinguish objects of K I E by the sign , and those of Elk by the sign  (which will not be confused with the fe5idue class field). We would write .
=
{} =
iJi9.
Let K/k be normal with group G. We define a descending sequence of groups C:: VI :J Va :J VI ::J V2 :> ... as follows. \Ii is the group of all 17 E G such that 17a
== a (mod !p;+I)
for all cr E O. " We also wrIte this condition symbolically (17 
1).0 C !pHI.
~ is called the ith ramification group of Klk, Vi and \ii+1 are not necessarily distinct. Vo is the inertia group and its fixed field T is the maximal unramified aubfield of K. We note that the groups \Ii are normal in G. Indeed, let a E V;, rE G. Then .,.17T 1a
 a
= T(f7T l cr 
.,.la) E
.,.\lJi+l
='.JJ'+l
and hence E V;, thereby showing that V; is DOl'Dlal. Suppose that the powers of a are a minimal basis: D = E 001'. Then obviously 17 E V; if and only if (0' 1)a E ~+1. This shows that V; = 1 for large i. If K/k is completely ramified, then {nil} is a minimal basis and therefore a E \ii if and only if un == IT (mod !pi+l). The different is given in terms of the ramification groups as follows: TaT 1
lSuch elements Q exist. If fj is an element of 0 whose resid~ class generates the residue class field extension and n is a prime element of .0. then either fJ or fj + n will be such an a. See
[16, IU, § 1, Prop. 3].
XI. HIGHER RAMIFICATION THEORY THEOREM
1. Let Klk be a nonnal extension. Then 00
ordK {}
= L(#l'. 
1) .
•=0 PROOF. If (} is a minimal basis, and I(x) the irreducible equation for (} over k, then {} = f'(a) = 1117;il (uo:  0:). For u E Vi but a f/. Vi+l, we have by definition ordK(uaa) = HI. It follows that ordK a = Z::'o(H1)(#Vi#Vi+l) and this formula. is easily seen to be equal to the one of the theorem, for instance as follows. Suppose # 'Vt = 1. Then our sum is equal to
(#Vo  #Vl) + 2(#Vl  #V2 ) + 3(#V2  #Va) + ... +t(#'Vtl  #Vi) = #Vo + #Vl + ... + #\.'tl  t#'Vt = =(#Vo 1) + (#V1  1) + (#V2  1) +. ,. + (#'Vtl 1)
=
00
o
= L(#'Vo 1). 0=0
We shall now derive some Ill!efulstructure theorems ooocerwng the ramification groups. We begiD. by remarking that we may assume Klk completely ramified without loss of generality. Indeed, we have THEOREM 2. Let K/k be normal with group G. ut E be an intermediate field fixed under H. Let V; be the ramifiC4tion groups of K/E. Then ~ = Vi nH.
PROOF.
o
Clear.
If Elk is llOrmal it is much more difficult to determine the relationship between the ramification groups Yt of E/k, and those of K/k. The next section will be devoted to this enterprise. For the moment, we stay in one field. Denote by k; the multiplicative group of units of k which are == 1 (mod pi) for i > O. Complete this definition by letting ko be the group of units of k, and kl k* itself. We ha.ve
=
kl :::> A4l :::> k1 :::>
~ :::> ....
Let K/k be normal with group G. The factor group Kol K. is naturally isomorphic with K*, the multiplicative group of the residue class field. If K/k is completely ramified, then K k and G operates with trivial action. Let K/k be completely ramified, Yo = G. We have
=
=
u E '" all TI (mod ~'+1) TI"l ;!!; 1 (mod !.P')
TI"l E Ki' ln fact, we see that TIO'l E Zl(Vo, Ko} is a unit cocYc1e. THEOREM 3. Let K/k be completely ramified. VO/V1 is isomorphic to a finite subgroup 01 K* (multiplicative), and is cyclic. Its order eo is prime to p.
1. WGHER RAMIFICATION GROUPS
PROOF. The map 11  ITa1 (mod q3) is Ii homomorphism of Yo into the multiplicative group of the residue class field, because rr u  1 is a lcocycle with trivial action in Ko/ Kl = K*. The kernel is precisely Vii and this proves that Yo/VI is isomorphic to a subgroup of K k. A finite subgroup of the multiplicative group of a field is cyclic, so Yo/VI is cyclic. If eo is its order, then we must have eo prime to the characteristic of k, as was to be shown. 0
=
THEOREM 4. Let K/k be completely ramified, i ;;i! 1. If p == 0, then VI
= 1.
If pI 0, then Vt/V;+l is isomorphic to an additive subgroup of K, and is of type (p,p, ... ,p). Vi is conse£fUently a pgroup. PROOF. Again the map 11  f no 1 (mod K'+l) for 11 E Vi is a l.cocycle in Ki/K.+l. Since Vi has trivial action on K i • it is a. homomorphism, with kernel V.+l. It follows that Vi/V;+1 is isomorphic to a subgroup of K = k (additive)
because the map X H 1 + TIIX induces an isomorphism of the additive group of the residue field and the multiplicative group K.f K,+I' H P = 0 then the group is trivial. If p > 0 then every element hllll period p, and since V./Vi+l is a finite group, it is of type (p,p, ... ,p), as was to be shown. 0 THEOREM 5. Let T E Yo and let C1 E V; (i ;;i! 1). Then TOT1CT 1 E Vt+l il and only if either C1 E V;+ 1, or T' E Vi. PROOF. We may clearly assume that Klk is completely ramified. = eIT, where e E Ko. Let
Let TTI
CTIT == TI + ,8TIi+l
(mod q3i+2),
where fJ ED. Since u E Vo, we have tre == e (mod \Jli+l). Hence CTTTI = CT(cIT) :; ella (mod q3i+2)
== en + e,8llHI e
(mod $'12)
en + (cTI)i+l,Bei
(mod \l.l'+2).
On the other hand, T,8 == {3 (mod !ll) because K is completely ramified. Hence
+ ,81" (rII)l+l :; ell + {3(cTI)i+l
TtrIT;;: TIT
(mod $1+ 2 )
(mod \Jli+2).
Combining the two congruences we get
(UT  ru)TI:; (ell)i+l{3(gi 1) (mod \)Ji+2). Replacing IT by (T1q11I) we get
n
TCTr 1C1 1 TI == (gTI)i+1 p(e'  1)
We~that Tl1r 1tr 1
E Vt+1 ~ (3(e i

(mod q3I+2).
1) = 0 (mod '.lJ)
~ ,8
== 0 (mod q3) or gi == 1 (mod \J.l) But from the definition of 11, we clearly have CT E Vo+1 if and owy if fJ:; 0 (mod q3). Since K is completely ramified, TC == g (mod \Jl) and therefore TOIT :;
This proves that our theorem.
7' E Vl
giTI (mod \)32).
if and only if ei
;;:
1 (mod q3) and concludes the proof of 0
,. XL HIGHER RAMIFICATION THEORY
COROLLARY
1. The group Vi/Vi+l is contained in the ~ter O/Vl/v,,+1'
PROOF. If we take corollary. COROLLARY
T
in VI then we must have
'TU'T1uJ
E
Vi+l.
This is the 0
2. If K/k is abelian, and Vi Sed in the preceding section. Such a field will be called a general local field. We note that aU finite extensions of k are solvable. Indeed, we need only consider normal extensions. Such an extension is a tower whose first step is unramified, and hence cyclic by Lemma 3. The second step is tamely ramified and can be obtained by adjoining an nth root. The final step is strongly ramified, and its group is a. pgroup which is cousequently solvable. LEMMA 7. Let k be a general local field and let K/k be an unramified extension. Then every unit of k is norm from K.
PROOF. The proof is based on the following result, valid in complete fields. We use N for the norm and S for the trace.
' GENERAL LOCAL GLASS FIELD THEORY
PROPOSITION. Let k be complete under any discrete tlaluation and let Kjk be a normal unromified extension. Let u be a unit in k, u == 1 (mod p). Then u = Ne, where e: is a unit of K. PROOF. Since Kjk is unramified, the prime p remains prime in K. An element of order 1 in k also h& order 1 in K. We show that given a unit U m == 1 (mod pm), there exists a unit em of K such that em == 1 (mod pm) and u,.,. == Ne: m (mod pm+l). Put e = 1 + ?imYm where Yrn is some integer of K to be determined. Write U m = 1 + ?imam where Cl: m is an integer of k. We have 11"
N(l + ?i"'Ym)
"=
II(1 +'II'mu:,) == I + 'II'ms(Ym)
(mod prn+l).
11'
We may interpret the trace in the residue cla:>s field. K corresponds to a normal extension Ko of the residue class field /CO, and not all elements of Ko have trace O. Since the trace is homogeneous with respect to the elements of ko we can obviously find an integer Ym such that S(Ym} == Cl:m (mod p). For this value of Ym we get
Nem == 1 + 7I'mS(Ym) == 1 + ?imam (mod pm+l) contended. We have u == NeI (mod p2). Put u/Nel = U2' We can find a unit e2 == 1 (mod p2) such that 'U2 == NC2 (mod p3). Put U2/Ne2 = "3. Proceed inductively in this manner. The infinite product elE2E3'" converges to a unit of K, and clearly u Ne, as w& to be ShOWll. 0
88
=
Returning to our general local fields, we let u be allY unit of k. According to Lemma 4, there exists a unit eo of K BUch that u == Ne:o (mod p). By the propOSition, u/Neo = Nel for some unit el in K. It follows that u = Ne where f; = EOe:l, thereby proving Lemma 7. 0 The next lemmas are preparatory to the proof of the second inequality, and to
the determination of the conductor in cyclic extensions of prime degree. If K / k is a normal extension, II denote;; a prime of K. ?i a prime of k, and an element of order ~ s in K. We use N for the norm, and S for the trace.
n.
LEMMA 8. Let K/k be cyclic of prime degree t. Let x be an tnteger 0/ k. Then
N(l + xlI,} == 1 + xS(II.) + Xl NIl. PROOF.
(mod S(ql2.».
Let 0' generate the group of K/k. We have
N(l + xII,) = (1
+ xII.)(1 + xlI:)(l + xlI~3) ... (1 + xn:'I).
Beside the three terms in the expansion given above, we have terms of the type x"rr:(u) where rp(O'} is a polynomial in 0' of degree ,..; 1, with coefficients either 0 or 1 and involving at least two mOIlomials. Furthermore all such polynomials will actually occur as exponents of lIs. If rp(a) = O'rp(O') , then rp(a) = 0'2rp(0') == crrp(O') = ... and hence rp(O') = 1 + 0' + 0'2 + ... + 0'£1. This rp(O') would give the norm, already taken care of. We have therefore 'P(O') :F O'rp(O'). We lump together the terms involving 0 then K/k is strongly ramified and l p. Since k is perfect, the index is 1. This settles the case i O. We shall now treat the cases where'i ~ 1. We distinguish several values of t, and use Lemma 9 constantly, m (t + l)(e 1). t 1. We have ¢(i) i. Therefore SqltP(i) pi. Also, we see that ~~(i) and ~(i)+1 are contained in pi+l We have trivia.Ily N\j:rI'(i) = pti C p'+l. Hence by Lemma 8 we have
=
=
=
=
=
=
N(l + ql",(i) == 1 +pi
(mod f)i+l)
which gives NK"'(i) . ki+1 = k., as was to be shown. t ~ 0 and i ~ t. We have ¢(i) t + l(i  t) = (t  l)t + ii. Therefore "'(i) + (t + 1)(l 1) ti + (£  1). From this and Lemma 9 it follows that
=
=
SlP",(i)
Furthermore,
SlP211>(i)
= p'
and
SlPI/;(il+l
= pHl.
C pi+!. Since t/J(i) ~ i, we have N!,JrI'(i) C p'
and
Nql",(i)+1 C
pHI,
The inclusion statements follows from these remarks and Lemma 8. To get the index we consider two cases. i > t. Then 1j>(i) > i. and NlIl'p(i) C pi Hence
N(l
+ ~(i»
=: 1 + pi
(mod pHl)
and "'+tNK",(.) = ko. i = t. We have already considered i = 0, 60 that we may assume i = t > O. We are in strong ramification. We have ,p(i) = i = t. By the inclusion statements, there exists an element Q E qlt, Q ;. lilt+! such that Sa ¢ plH and Sa = brr t ,
Xl. HIGHER RAMlFlCATION THEORY
b ¢ p. Then No == 71'(a, where a t 0 (mod pl. For any integer x of k we have by LemmaS: N(l
+ xo) == 1 + xSa + 7;1 No (mod pHI) E 1 + '1I't{axl + bx) (mod pHI)
where ab ¢ 0 (mod p). We have considered the polynomial I(x)' = axl + bx in Lemma. 6, and found the index (k : f(k» ~ p. Multiplicatively, this gives (k t : kt+1NKd ~ p, as was to be shown. t~Oandi i+(i+l)(tl) = li+ll. This proves that !./J{i)+(t+l)(l'I) ~ l(i+l). Hence by Lemma 9, SI.P"Ci) C pHI, and a fortiori, ~(i)+l and S~2"'(i) C pHI. For the norm, we are in the ramified case, and therefore N!p"'(i) pi, and N':P"'(i)+1 pHI. We have !./J(i) == i
=
=
Using Lemma 8 proves the inclusion l;tatements. Furthermore, N(l
+ \13>/1(;» = 1 + p'
and this me8.DS multiplicatively ~+lNK1/J(i) == our theorem.
(mod pi+1) ~,
thereby completing the proof of 0
Having settled the case of a cyclic extension of prime degree, we can treat the general normal extension by showing that the statements we wish to make are transitive. We recall the notation: ~(x) and tbHx) are the right and left derivatives of 1jI(x). We denote by \f.>~/f(x) the q?otient 1jI~(:t)N~(x). THEOREM 9. ~ 1
an integer i
Let k be a general local field, and Klk a normal extension. For we have:
1. NK",(i) C k., and NK.p(i)+1 C 2. (k;: ki+1N K",(i») ~ "'~/l(i).
kstl'
PROOF. The two statements have been proved in Theorem 8 in a cyclic extension of prime degree. We know that Kjk is solvable, and it suffices therefore to prove that the two statements are transitive. This transitivity is essentially a trivial consequence of the transitivity of the nonn, and of the 'I/J function (proved in Theorem 7). Let K ::> E ::> k be two normal. extensiOllS and assume the theorem for K IE and Elk. We have
By assumption, NE¢(i) C '"
NE~(j)+1
C
kHI
and NK;j;(~(i» C and
E;ji(i)'
NK~(~(i»+l c E~(i}+l'
The first statement is DOW trivial, because NK,,(i)
= NNK",(.) C NEiJ(i}'
To prove the second statement, we write
(/0.;: ki+INKtjJ(j» = (k.;: "'+lNE~(i»)(k.;+1NE~('l : ~+INNKJ(~(i)))'
4. GENERAL LOCAL CLASS FIELD THEORY
We can insert the group E~(i)+l in the last index because of the inclusion N E~(i)+l C kHl whim we have just proved:
I\;+lN[E,ji(i)+1NK~(~(i»] ... l\;+lNNK~(..p(i}l· Hence our index becomes by induction
Et if;~/l(i)(Eifo(i) : E~(i)+lNK~(~(i))) ~ 1fr~/l(i)~/t(if;(i)) = 1/J~/l(i). This final :3tep follows from the chain rule for right and left differentiation which bolds because our functions are strictly monotone. 0
If we take i very large in the preceding theorem, we get ~/t(i) =< 1. This means that for all sufficiently large i, (I\; : k;+lNKtfJ(i)}
=1
and from this index we see that every unit of ki iB congruent to a norm from KtfJ{i} mod PHI' Such a Wlit can therefore be refined to a norm by all obvious argument. COROLLARY. Let k be a general local field. Let K/k be a normal extension. Then k; C N K for surne integer i.
The conductor of K/k iB the least power p" of P such that k" c NK, and we see that the conductor exists. We denote it by f K/,,, or briefly by f if the reference to the field is clear. We may now write the norm index as '" product:
(k: NK) = (k: koNK)(koNK: kiNK) .. . (k._lNK : k.NK). If IS is big enough, k. C N K. Hence DO
(1)
(k: NK)
= II(kjNK: ki+lNK). 1
Furthermore, by an elementary isomorphism theorem, we have (2) (koNK: I\;+1NK)(~ n NK: ~+lNK",(O n NK) = (I\; : ki+lNK1/>(i»
:;;;; tP~/£(i) (by Theorem 9).
From (1) and (2) we obto.in (3)
(k : NK) ~
co
IT t/!~/t(i) Et n 1
where n ;;: [K : k]. Indeed 'I/I~(i) ~ ~(i + 1) whence l&~)l) ~ 1. t/!Hl) = 1/1, and .,p~(oo) = e. Hence the product is"; ef = n.
FUrthennore
The index inequality of (3) is the second inequality. The claBS field theory in general local fields may now be developed in exactly the same way as the theory in the classical case. k; a consequence of the second inequality, we can prove that hz(K/k) ~ [K : kj. and then use the same method as that of Chapter XIV. Instead of the FrobeniU::l Substitution, we select in the Galois group of the algebraic closure of k an element it whim operatea non trivially on every finite extension of
XI. HIGHER RAMlFICATION THEORY
98
k. The effect of it on each finite extension is to generate the. Galois group, which is cyclic. The automorphism (j has a corresponding a.utomorphism a in the Galois group of the maximal unramified extension of k, which can play the same role as the Fl'obenius Substitution. It has not been canonically selected, but this does not make any difference to the proofs, as long as we develop only a local theory. It is only in the global theory, when the local theories were pieced together, that it became important to choose the proper automorphism in the local fields. Having done local class field theory, we obtain all the results of Chapter XIV. In particular: (0
THEOREM 10. Let k be a general local field. Let n be its algebraic closure, and the Galois group. Then (~,n·) is a class formation.
We may now return to the indices computed to prove the second inequality. We know that in abelian extensions the norm index is equal to the degree. Thls means that the doubtful indices of Theorem 8, 9 and (2), (3) above for abelian fields are no more doubtful, but are actually equal. THEOREM 11. Let k be a general local field, and let K/k be a finite abelian extension. 1. ~ ("I. NK = k'+lNK.,i) n NK c ki+1NK",(i). 2. (ki : k'+1N K,p(j») == (kiNK: ki+lNK) == ""~/t(i). 3. All breaks in'l/.>(x) occur at integral arguments. In other worr.b, Vf../l(x) "" 1 if x is not an integer. 4. kz C NK if and only iIV~(=.V,p{:t» ::: 1.
PROOF. Statements 1 and 2 arise from the equality in (2) and (3) above. 'Th prove 3 we note that yp(x) has a finite nUIIlber of breaks. If we take the product n'l/.>;/l(x) over the numbers :r: for whlch there is a break, we get ef = n. But ~ ¢~/l(x) is > 1. Since the product over integers already yields n, this means that the integers give all the breaks. Using the equality of the indices in 2 we have:
k.. C NK
~ (k j : kHiN K,p(j» '= 1 for all j l/l~/l(j) = 1 for all j ~ 2:
~ 2:
'I/.>~(x) = ypHx) = yp'(oo) = e (#Vo : 4W",{o:)} = (#Vo : #V".,) V,p(z) = V"" = 1 VO: = 1
o
thereby proving 4.
THEOREM 12. Let k be a general local field, and w the norm residue mapping, into the Galois group 1!5 of the maximal abelian utension of k. Then w(k.,) is everywhere dense in !?)x.
PROOF.
Let K/k be a finite abelian extension. K is left fixed by
=1 k., c NK w(k",) leaves K fixed
!l)'" Vi/I
E::> k be two normal extensions. Let X be a character of GI H = G, so that X may be viewed as a character X of G also. Then
II(X) :::= II(X). PROOF. Let = Uc'7(vt n H). We know that Vt =: VtH/H, and the it are na.turally elements of V t , and X(u) = X(u'Y) for all 'Y E H. It suffices therefore to show that
Vt
f X(lu)do'= ~ X(lu)dO'. lv· lv·
5. THE CONDUCTOR'
Note that #vt == #Vt· #(yt
l"X(lO')da=
101
n H), and hence
EVi;~tO') = #(~~H)~X(1_iT)= lv_.:WiT)da.
0
v' THEOREM
17. v(x)
PROOF. Since
yt =
= ~E~oEv.x(IO'). we have
V"'(t),
v(X) = .
1
00 ((
Jv,,'t)
1
X(I  0')00) dt.
The function f(t) "" J,v..".) X(l 0)00 has only a finite number of discontinuities, and we may take the sum of the integrals in the intervals where no discontinuity occurs. Let 8 = ¢(t) and t = = Vi+! for 0 < 0 < 1, &ncl
l'
'+1 (
2: X(I  (1) )
ds
= 2: X(l 
(1).
o
v,+l
V.
The formula giving veX) in the preceding theorem might have been taken as a definition. We have selected the integral because the important Theorems 13, 14, 15, and 16 were immediate consequences of our definition. THEOREM 18. Let K/k be norma.l with group
G. Let H be a subgroup, and E
the fixed field. Let ¢ be a character of H, and X the induced character of G. Then
PROOF.
h: = D~W NE/kf",. (D = discriminant.) Let G = UHe. The induced character X has the value X(T) =
L¢(crc 1 ) c
where ¢(11) = 0 if 11 ¢ H. Starting with the expression derived for veX} in Theorem 17 we have therefore:
XI. HIGHER RAMIFICATION THEORY
102
But ~fl =
Vi because Vi is normal. We can therefore take ~ut the c 1: =
1
00
1
00
eLLLtP(lo) o v."
.
= eLLL 1/;(lo) o
c
V;
"" = 1 L(efLW(lo»). e
Since e =
ee and 1/;(a} = 0 unless 0 E Vi n H,
= ! f: (#ViW(I) e
But
v,
0
V. n H =~.
1
#(Vi n H)1JJ(I) +
0
L
,p(1 
a»).
V;nH
We may add 1 and subtract. 1, thereby giving
1
00
eo
LL ,p(10').
1""
00
"" ; L(#Vi 1)1JJ(1} + ; ~)#v. 1)1/1(1) + f;eo
e O j i,
Using Theorem 1 fur the order of the different, and Theorem 17 giving the expression for 11(1/;):
= !1f;(l)[ord K !J ordKD] + jlJ('I/J). e =
But ordK!J  ordK'; = ord K 3. Also, lordK ordE. Furthermore, ordkNe/k = jordE . Combining these remarks, and writing the formula multiplica.tively, we have
fz: = p"(x) where Delle
= D~WNEI"f",
= N E /,;;; is the discriminant.
This proves our theorem.
0
=
Ilf'e
COROLLARY 1. Let,p 1 and let the induced character X == E; Jl.sXi where Xi the irreducible characters 0/ G. Then
DE/i: PROOF.
= fx == II ~.
Imw.edia.te from the additivity of the characters.
COROLLARY
o
2. Let K/k be abelian. Then DK/k
= ITfx x
where the product is taken over all the ordinary characters o/G. PROOF. Let E = K in the preceding corollaxy. The character X induced by 1 is the character of the regular representation. The irreducible characters have dimension 1 50 ~i = 1 in the product. The irreducible characters are simply the ordinary characters of G when G is abelian, and this proves our corollary. 0
The preceding theorems giving the formalism of &I(X) are valid in a complete field with any perfect residue class field. We shall now specialize to a general local field to obtain one more statement concerning the conductor:
103
Ii. THB CONPUCTOR
THEOREM 19. Let k be a generalloc4ljield, and K/k a normal extension. Then v(X) is an intege,.. PROOF. A character is a. linear combination with integer coefficients of irr~ ducible characters. Using the linearity of v(X) we see that it suffices to prove the theorem for irreducible characters. Suppose next that X is Idimeosional. Such a character will be called more briefly an abelian character. It is a homomorphism of G, and its kernel Go is a normal subgroup. FUrthermore, G = GIGo is cyclic. Let X = X on G. Then II(X) = D(X) by Theorem 16, and by Theorem 13 we know that v(X) is an integer. This proves the theorem if X is abelian. Let 'I/J be an a.belian character of a subgroup H. Then iI(fjJ) is an integer by the preceding remarki>, and f", is integral. Let X be the induced character. The formula
f)C = D~W NEllcf", of Theorem IS shows that h: is also integral, i.e. that v(X) is an integer. Brauer has proved that every character is a linear combination with integer coefficients of characters induced by abelian characters. In view of the linearity of v(X) it follows that v(X) is an integer. We shall give here a proof independent of Brauer's Theorem, by reducing the problem to pgroups. Indeed, for a pgroup, it is shown fairly easily that every irreducible character is induced by an irreducible abelian character, 2 and the preceding argument can then be applied directly. We may of course assume Kjk completely ramified. e == eopr = #Vo. We must show that e divides E.':o Ev. X(I : u). We begin by treating eo. Let '(X) = ev(X) be the double Bum. Then OQ
'(X)
= E[X(l)#Vj  Ex(u)] Yo
;=0 OQ
=L:>[X(l)#V.  LX(u)] 
.=0
Vj
OQ
~)il)[x(l)#Vj  ~x(O')] V
;=0
= Ei[X(l)#Vi  Ex(u)]  ~ i[X(l)#V;+l E x(u)] 00
""
V,
i=l
v.+ 1
i=1
"" =X(I)#Vo LX(U) + Li[x(l)(#Vi#V;+l)Vo
;=1
L
xCu)].
V.V'+1
But eo I #Vo and ~v. X(u) = #Va· ~ where ~ is the multiplicity of 1 in X. SO the first term is divisible ~>y eo. To handle the sums, we recall Theorem 5 and the discussion following it. The sum EV;VHl X(u) breaks up into a sum over equivalence classes and we have
L \';V.+l
xCv)
= l: ,..,X(u) f1
where the second sum is taken over a representing the equivalence classes. (We can do this because a character depends only on the conjugate class of a group element. ) ~See th~ corollary of Theorem 3 in the appendix which
follows.
104
XI. HIGHER RAMIFICATION THEORY
We know tha.t EH x(a) is a rational integer for any subgroup H of G. Hence ~v.. '" . .. + 1 X(o') = • X(o') ,+1 X(U) is II. rational integer, sa.y mi' We also
v..
know that
Lv..
eo I iTa.
Ev..
Hence
'
ms = eo ~)iTC7/eo)x(O').
=
But L.,.(ira/eo)X(O) mo/eo is an algebraic integer, and a rational number. It is therefore an integer, and this proves that eo divides the questionable sum. The preceding discussion is valid for any character X. Taking X = 1 yields the term i(#Vi  #Vi+1) which is thererore also divisible by eo. All that remains to be shown is that veX) is divisible by p". We have >.(X) = 2:=l 2:v, x(l a) + I:vo X(l 0'). The sum 2:vo X(l a) is == Vi n Vl divisible by #Vo ;: e. Let E be the fixed field of VI. We know that and therefore
Vi
00
00
LLx(lO') = LLx(lO') i=1 V,
i=l
where Xis the restriction of X to Vl. We have in Eby
V.
Vo = Vt.
Our sum differs from l:(X)
and this sum is divisible by #Vi = pro Hence it suffices to prove tha.t X(X) is divisible by p' and reduces therefore the theorem to the case of the pgroup Vl. We saw already at the beginning of our proof that the theorem follows in full, if we use the fact that every character of a pgroup is induced by an abelian character of a subgroup. 0
This concludes our discussion of the higher ramification in general local fields.
Appendix: Induced Characters For the convenience of the reader we shall develop here the theory of induced cha.ra.cters, used in the preceding section. We begin by recalling basic notation. Let G be a finite group, F a field, algebraically closed and of characteristic O. All the spaces we deal with will be finite dimensional Fspaces. This will not be mentioned explicitly any more. We denote Fspaces by U, V, W•.... Let V be a Gspace. G acts on V linearly, and is represented by linear transformations of V. If a basis of V is selected, then the trlWSformations may be given by matrices and we have a homomorphism
a .... M a of G into a group of matrices. The character X of the representation is the function on G given by
x(O')
= SCM,,)
where S is the trace (sum of diagonal terms). The fundamental theorem states that the character is an invariant of the representation, and that in fact it characterizes
APPENDIX: INDUCED CHARACTERS
105
it: Two representations of G are defined to be equivalent if their spaces are Gisomorphic, and two representations are equivalent if and only if they have the same character.
Induced representation. Let H be a subgroup of G. For each coset c = H 0', let c be a representative element, so that G = U c 1H, disjoint union. Let V be a Gspa.ce, Wand Hspace, and i : W + V an Hhomomorphism. One says that V is induced by W via i if i is an isomorphism into and V :;: $c 1iW, direct Sum. For each W there does exist such a V, for example, V = F[G] ®F[H] W, with i(w) = 1 ® w. For any such V it is easy to check that if V' is another Gspace the map J t+ f Q i is an isomorphism Homo (V, V') ~ HomH(W, V'). From this it follows that V is uniquely determined by W up to a unique Gisomorphism f such that f 0 i = i'. Therefore we usually view i as an inclusion, don't mention it, and write simply V
= Vw.
THEOREM 1. Let G:J H::> I, and let IV be an Ispace. Let Uw be the Gspace induced by Wand Vw the H space induced by W. Then Uw ~ UVw' [IWl and W2 are two Ispaces, then Vw,+w. ~ Vw, + Vw. where + mt4n.s the
direct sum. PROOF. The theorem is an immediate COJI8equence of the Wliquenees of the Wduced representation. 0
Let G :J H, and let W be a.n H space. Let t/J be the character of the repre&entation of H in W. The character X of Vw is called the induced character. The next theorem gives a formula which allows us to compute the values of X in terms of the values of t/J. THEOREM
2. Let G
= UHc.
Then
X(u) ""
:E ""(CdC
I)
c
where tve let 1/;(7")
=0 unless l' E H.
PROOF. Let {1"",{n be a basis for W oveI F. We know that V = Let u be an element of G. The elements {cO'l{dc,i form a basis for V.
Remark that 0'(CiT 1{) = Cl(cuCO'l)~.
The action of 0' on this basis is therefore given by u(w1e.)
= c 1 ~)CaCo'I)1,.e,.
,.
= L(Co'CQ'1)iU{C 1 e,.).
,.
By definition,
But
CO'
= c if and only jf 001: 1 E H. t/J(wc 1)
Furthennore, = L(CdC 1) ••• i
Ec 1W.
106
XI. HIGHER RAMIFICATION THEORY
X(O') ==
L v(eoe
l)
(:
o
as was to be shown.
THEOREM 3. Let G be a pgroup. Let V be an irreducible Gspace. If V is not Idimensional then there exists a proper stlbgroup H1 and an irreducible H1subspace W of V such that (G, V) is induced by (H1' W). PROOF. We begin by recalling that an irreducible representation of an abelian group is Idimensional. This implies in particular that if V is not Idimensional, then G is not abelian. We shall first give the proof of our theorem under the additional assumption tha.t V gives a faithful representation of G. (This means, if O'e = { for EV then 0' = 1.) It will be easy to remove this restriction at the end. Let H be a normal subgroup of G which is abelian and contains the center properly. (Proof of existence of H: G has a nontrivial center Go. Let C == G/Go. Let 11 be an element of period p in the nontrivial center of (j and let fI be the subgroup of G generated by U. Then fI is normal in G. Its inverse image H in the natural map G + (j is normal in G, and is generated by an inverse image 0' of ij, and by Go. Furthermore, 0' '/:. Go so H ::) Go properly. Finally, 0' commutes with its powers and with Go (the center of G) 50 that H is abelian.) We denote the elements of H by'Y. AE. an Hspace, V is a direct sum of irreducible Hspat;eS which are Idimensi
all,
anal.
Let,
E V generate a Idimensional Ht;paee. Let '"
be its character, so 'i'e =
tP(r)e. where Ibb) E F. If 11 gives an equivalent representation, then "'If! == "'bY"I.
If a, b E F then 'Y(a{ + 1Tq)
"'b)(a{ + brj}. Hence the vectors of V giving rise to the same irreducible representation of H form an H space W. We contend that V i W. Assume V == W. Let e be any element of W V. Let 0' E G. Then 0'1 t is a Idimensional Hspace by assumption, and has character f/J.
=
=
Hence 'Y(a1t)
= "'("'I)O'l~
= uf/J("'I)u 1e;:;: "'('Y)e. This shows that y 1. Furthermore, (H), WI) is a local constituent of (G, V). We contend that WI is irreducible for H). This will finish the proof of our theorem (in the case that V is faithful). Suppose WI is not irreducible for HI' Then there exists a. space U C Wi> U i= WI such that H 1 U c U. Let G UHlc. We know that V = L c C I W1 • Let V' L c1U. Then VI f:;. V, and we shall prove that V' is a Gspace. thereby contradicting the irreducibility of V.
=
=
We have
uarlu = C 1(wcu I )U c ciU because (wenI) E HI. But CO'runs through all cc.x:>ets as c does, and this proves that uY' c V'. Hence V' is a Gspace, contradiction. Suppose now that V is not faithful. Let Go be the normal subgroup of G con.isting of all u such that == ~. for all { E V. Then V is an irreducible space of = GIGo and gives a faithful representation of If V is not IdimeIlllional, then G is not abelian and there exists a proper subgroup H of (; and an irreducible fIspace W such that (fI, W) induces (a, V). Let H be the inverse image of if in the natural map G + G. Then H J Go. and W is naturally an irreducible Hspace. FUrthermore we contend that the stabilizer HI of W is H. Certainly, HI :::) H. Suppose u E Hlo a f/ H, then uW C W. iJ ¢ fl. Since a has the same effect as u on W, this coutradicts the fact that iI is the stabilizer of W. This proves that (H, W) induces (G, V) and concludes the proof of our theorem.
a
ue
a.
o
COROLLARY. Let G be a pgroup. Let X be an irreducible character. If X is not Idimensional then X is induced by a Idimensional character.,p 0/ a su.bgrou.p H o/G. PROOF. Let V be the irreducible space of x. A subgroup of a pgroup is a pgroup. We may apply the preceding result step by step using Theorem 1 Wltil we cot a. subspace W which is Idimensional. The character 1{J of W will then induce
X.
0
CHAPTER XII
Explicit Reciprocity Laws 1. Formalism of the Power Residue Symbol The global norm residue symbol was obtained from local ones. The definition of the local symbol was obtained indirectly from a nonconstructive proof of the fact that all cocycles have an unrarnified splitting field. We are therefore faced with the following unsolved problem: To determine explicitly the effect of the norm residue symbol on totaJIy ramified extensions. (On the unramified extensions, it is the FrobenilUi Substitution.) We shall treat special cases of Kummer extensions. We need an auxiliary algebraic statement. LEMMA 1. Let F be a field of characteristic p ~ 0, containing the nth roots of unity, P f n. Let a E F. Then 0 and 1  0 are norms from F( a l/n ).
PROOF. We distinguish two cases. [F(a l/ ") : F] = n. Then x"  a is irreducible and its roots generate F(a 1/"). We have . x"  0 = (x  (nol/n)
n C,.
where Cn ranges over all nth roots of unity, and (n0: 1/" are the conjugates of olIn. Putting x = 0 shows a is a. norm. Putting X = 1 shows 1  a is a. norm. [F(a l !") : F] = d < n. By Kummer theory, din. Let ct"· = f3" where din is the period of a (mod Fn). We have F(ol/n) = F«(31/d) and by selecting a.1l n and (3l/d. suitably (Le. by multiplying them by a suitable root of unity) we may assume a lln = f31/d.. The conjugates of al/" are therefore (d.al/n where (t coefficient divisible by 1f exactly, we have, for suitable do
J(t)  clo!.p(t) = bIt + ~t2 + ~t3 + ... and 2::'1 b"t" is in the kernel. Hence by the same argument we can find d1 E ., such that
88 before, 11' I b1and
J(t)  (do + dlt)!.p(t)
= c2f + cst3 + .... Repeating this argument shows that I(t) = g(t)",(t) where get) tended. Let a E 0, a derivative,
= I(ll) == E:'oa"n" a' :;
E oft}, as con
where I(t) E oft}. We may take the
L llaoIT,,l.
Then a ' is not well defined in K. However, we know that any other expression for a as a power series is of type a =0 J(II) + g{II)tp(II) where get) E o{t}. Taking the derivative shows that 0:'
== t(rI) + g(rI)tp'(ll).
But O. But
= (n 
ordx"jn ordx
l)ordz  ordn
'> (n l)/(P  1) ==
> ordx for n ~ 2, or
r
(because n ~ 2)
«n 1)  r(p l»)/(P 
1)
~O.
To justify this last step, note that it suffices to show r(p  1) ~ n  1. Since p" ~ n it suffices to show r(p  1) ~ p"  1. If r = 0 this is clear. Otherwise, dividing by
P  1 shows that the inequality is equivalent to r ~ 1 + p + ... +prl, which is true because there are r terms on the right. This concludes our proof because of the strict inequality in the second step. 0 REMARK.
If 0;;;; 1 (mod p) we define log a
THEOREM 2.
Let a and {3 be == 1
(mod
pl.
=log(l 
(1  a».
'Then
loga.8 = logo + log,B. PROOF. The identity in formal series is known, and all the series involved 0 converge by the preceding theorem. COROLLARY.
Let ( be a p. root of unify in Ie. Then log ( is defined and = O.
PROOF. We certainly have 1  , == 0 (mod p) aod hence log(l  {l converges. By the functional equation,
0= log 1 = log (p'
Heuce log (
(»
= p·log(.
=O.
o
Although we shall not need the exponential function in the sequel, we give it
here anyway [or completeness. THEOREM
3. The series
exp x = 1 + x + z2/2! + x3 /3! + ...
Jor all x 1l!JCh that orr! x> l/(P I). In that case, ordzfl./n! > ordx for 2, and ordx = ord(expx 1).
COtWe7ye8
n
~
PROOF.
Write n in the padic scale: n ao+alP+ ... +a.,.pr
2. LOCAL ANALYSIS
113
where all are rational integers, 0 :S;; au :S;; p  1. Then
[nip) = 0.1 + (J.2P + ... + (J,rp,.l [n/p2) = ~ + ... + ar P,.2
We clearly have p.
= oed n! = [nip) + In/p2) + ... + In/prj = =0.1 + (1+p)a2 + ... + (1 +p+'''+ p"l)ar
and therefore (P  l)p.
= (I 
1)0.0 + (p  I)al
+ (p2  1)0.2 + ... + (pr  1)a,.
=n8" where
8"
= Go + 0.1
+ ... + 0.,..
This gives
p.
Now
= (71 
sn)/(P  1).
ordx" 1711 = nordx  p.
=71 (ordx p~
1) +s,,/(Pl).
Since Sn is positive we see that ordx"'ln! > 00 when ordx > II(P  I), and the series converges. To get the second statement ~ have to show that ordx"/n!  ordx > O. We have ordxn/n!  ordx = (n l)ordx  (n  sn)/(P 1)
=(n I>(ordx  pl _1_) + (sn l)/W  1). If n ;?; 2 the result is obvious because Sn THEOREM
;::
1 always.
o
4. Ilordx and ord y > Ij(P  1) then exp(x + y) = (expx)(expy).
PROOF. The formal identity is known, and all the series converge by the previous theorem. 0 THEOREM
5. Ifordx> l/(Pl) then explog(1 + x) logexpx
= 1 +x
=x.
PROOF. The formal identities are known, aud all the series converge by the previous theorems. 0
Let" = [e/(pl)j+l. The log and exponent give mutually inverse isomorphisms of 1 + p6 onto pol. We leave it as an exercise to the reader to prove the converse of the Corollary to Theorem 2, i.c. the kernel of the map for
Q
!!!!
Q logo 1 (mod p) is exactly the p"th roots of unity contained in the field k.
XII, EXPLICIT RECIPROCITY LAWS
114
3. Computation of the Norm Residue Symbol in Certain Local Kununer Fields We treat the simplest cases of Kummer fields, and begin with the infinite prime. Let It be the reals, 0, b E lit. We may interpret the results of Section 1 with II = 2 over the reals. We obtain
(a, b) PolO _ (_ 1).ia".AI.'JD!1 .
e
Next we consider finite primes. Let be a prime and k == Qt((t) where Qt is ladic completion of the rationals and (t is a primitive fth root of unity. We shall determine (0:,11) in the field k({Jl/t) explicitly. Since the symbol is continuous and multiplicative in both arguments it will suffice to determine it for a multiplicative basis of k. We first consider t' = 2, k = Qz. A multiplicative baBis2 of (h is given by 2, ~e
1, and 5. THEOREM 6. Let
a,
U2 be ~nits (2,2) :;; 1
bE
of~.
Then
= (_1)(4 1)/8 (a, b) = (b,a) = (_l)e(4)c(b)
(2, a)
2
where c(a) ;: (a  1)/2 {mod 2}. PROOF. We see that (a,b) == (b,a) from the inversion theorem and the fact that (a, b) = ±1. We prove that the exponent c(a) is multiplica.tive. We have
al  02  1 ==
111a2 
(a1 
1)(Cl2  1) !!!! 0 (mod 4)
'.,
and therefore
Thus
e(ala2) = e(al) + e(a2) (mod 2).
For the other exponent we have (0102)2 
and therefore
o~ 
af 1 == (af l)(a~ 1) ;: 0
(mod 64)
(al a2)2  1 = ai  1 + ~  1 (mod 2) 8 8 B
as was to be shown. Both sides of OUI equations a.re multiplicative. It suffices to verify the sta.tements for a basis of k. We note that: Q2(i) is ramified (because (1 + &)2 = 2i). Cl2 (~) is unramified. Indeed, let 0 = ( J5  1)/2. Then 0 generates Q2 (vrs) and satisfies the equation 0 2 +0:1 = O. Read mod 2, it is the canonical equation for an extension of the residue class field. Hence Q2( Q) Q2 (v'fi) is unra.mified. Now:
=
2Here 8J\d in the next four pagea, "multipliC4tive ba8is for" meana ~a. set; of genera.tors for a d _ subgroup of" •
11Ii
3. COMPUTATION OF THE NORM RESIDUE SYMBOL
(2,2) = (1,2) 0: 1 becaUBe 1 = N(l + J2) is 8 norm. . (2,5) = 1 because the prime 2 is not a norm from the unramified extension 4~MJ5)· . (1.5) = 1 because ~he unit 1 is norm from Q2(.;5). (5,5) = 1 for the same reaBon. (1,1)::: 1 because 1 is not a. norm from fJ2(v'i). (Otherwise, 1 = x 2 +11 which is impossible mod 4.) One verifies directly that the values we have just found ooincide with (_l)£{a)t{b) in each case, and this proves our theorem. 0 We suppose from now OD that i is an odd prime and that k = Qt«") where, is primitive loth root of unity. Then (k : Qt] = e 1 and k is completely ramified over Qt. Let ). = 1(". Then ). is a prime in k, and e '" ).tl. It is actually easy to determine £1>.11 (mod >.). Namely: X I  1 + ... + 1 == n~:,~ (x  (") and therefore f = rr!:'~(I  (") We get II.
£/>. 11 = 1I(1(") = IT(I+(+ ... +("1) 11(1  ()
,,=1
.
But (== 1 (mod ).). By Wilson's theorem, we see that
£
>.11
== (£  I)! == 1 (mod >.).
We shall abbreviate the modulus, and write (>.) for (mod .>.). Let I1i = 1_>.i, t;;r: l. LEMMA 3. The 'Ii form a multipticative basis of the 'Units 0/ k == I().). A full multiplicative basis is given by the 11i, the (£  l)th roots of unity, and the powers of a prime. PROOF. The residue class field of k is Zt = Ql because k/Ql is completely ramified. The (£1) roots of unity lie in Gt and obviously any Wlit can be multiplied by such a root of unity to make if == 1(>.). Every unit e == 1(>') can be written e = 1 + alA + a2,\2 + ... where a" are rational integers. We can obviously solve formally
(1 +alt+a2t + ... ) = (l_t)0.1(1_ t 2 )b,(l t3)c3
for integers
b:.J,
•••
e3, by a recursive process. The power series obtained by putting 0
t = >.. will converge, and this proves our lemma.
We are now interested in computing the symbols ('Ii, >.). We note that
(1'],,>")
= (1 ).i,>.i)
0:
(1 ).i,>.)i
=1
by Properties 1 and 4 of the symbol. This shows that if because ('Ii, >.) is an loth root of unity. LEMMA
4. Let e be a unit of k, e
tfi
then (l1i,).)
= r().t+l) for ,orne:r E k.
=1
Then e iI an
ttiI. power in k. PROOF.
that
Suppose e =:; r().") with" ;;r: t
+ 1.
We try to refine x. We contend
116
XU. EXPLICIT RBClPROCITY LAWS
Indeed, the remaining terms of the binomial expansion are of type
f) ""
I (~) the intermediate terms are divisible by t ( >.2"1+1 and hence are O{>,II+1). Furthermore Alii It; >''' (>.,,(11)l(i1)). Also, III. ~ I, and £1 ~ 2. Hence the !.a.st term is also E O(N'+1). This proves that we can solve for y such
Since t E
that
xl + yx'  1 >'" == c( >. ..+1) and by a standard refinement prOce!lS, we can find an tth root for c, as contended.
o
THEOREM
7. For all i '" f. we have (1'Ji, >.) = l.
e
PROOF. If i < t then t i and we have already considered this case. If i ~ t + 1 then '11. == I(.\t+l) and "Ii .., (i by the lemma. Hence ("Ii, >.) (0, .\)l 1 thereby proving the theorem. 0
=
=
It will be much harder to prove the following results. THEOREM 8. ('1t,>')
= (1, or equivalently, (A,f'll) =(.
PROOF. The inversion follows f;om the inversion property for the symbol. We shall deal with (A, '111), because we enjoy the advantage of the following statement: k('1i/l)lk is unra.mifted. PROOF. Let A = fJ;/l. Then At  1 +).1 = O. Let B A = >'B + 1 and therefore B satisfies the equation
= (A 
1}/>.. Then
0= (>.8 + 1)1 1 + At
== >.1 Bf +
G)
>.11 B'1 +.... + tAB + A'.
But l"" )"tl. Divide the equation by ).t. Since
t I (~
we see that B satisfies
,.,t + (tA/>.l)Z + 1 == 0 (>.). We have seen a.t the beginning of our discussion that (lA/>.I) equation becomes therefore
E
1 (A) and our
Zl  X + 1 == 0 (>.). Since k/Qt is totally ramified, the residue class fields k a.nd Ql are equal and are simply the prime field Zt. Hence the equation above is irreducible in the residue 0 class field and this proves that k(A)/k is unramified. We conclude that u = (A, k{A)/k) = therefore prove that AI"l = (.
I(J
is the Frobenius Substitution. We must
3. COMPU'l'ATION OF THE NORM RESIDUE SYMBOL
We know that lJ'P on Awe have
== Bt (>'), and therefore B'P == B 1
11'1
(~). To get the effect
A'P 1 = (1 + >'B)'I' /(1 + >.B) = (1 + >'B'I')(I.\B + >.2&  ..• ) iii 1 + >'(B'I'  B) (>.2) i!! 1 _ >. (A2)
==,
(>.2).
But if(" = '" (>.2) then ("(1_("") =0 (>.2). Since (1,,,,,) "" >. unless (mod t) we conclude that A",,l = ( thereby proving our theorem.
II;;;
P 0
Our next problem is to determine (1]" f/j) in terms of ('Ii, >.). We begin by deriving a certain functional equation for ('1};,1/i)' We see immediately from the definition that
+ ),i 1]; = fli+j'
fli
=
Note that {1)l 1 because t is an odd prime, and hence (l.tJ) = 1 for all {3. From the addition theorem we get
(fli. >.11];)
= (flj, '1Ii+j)('1'+1, >.3'1;).
We use the wu]tlplicativity of the symbol and ('1i' >.i)
(1)
to get
= (1 
>.i. ),i)
=1
(fli' f/i) = ('1;', 'h+i )(f/i+j. '1/i)(1/i~i' A)i
and invert everything:
(fl., '111)
(2)
= ('1/i, flHj ){11i+,.. 1'Ii)(1/i+j. >.)i.
PROPOSITION.
('h, '1j) =
IT (17ri+'i. >,)(1'O
iHoi)
r,.~l
(r,.)=l
where r. s are positive integers, relatively prime, and for each pair (r. s) the pair (ro, so) is one solution of the equation TSo  STo = 1. PROOF. We must first show that tile expression is well defined, Le. does not depend on the choice of (ro, so). Indeed, it is easy to verify that any other solution (Tb sd is given by rl = TO + rt and S1 = So + st where t is an integral parameter. This implies that the exponent changes by (Ti + sj)t. Let 11 = ri + sj. From (1) we know that (1)1,2) tile obtain a(" == 1 {>,2). This reduces the computation o{(a,(J) to the case where Q == 1 (>,2), and to «,13) where {J == 1 (>.). REDUCTION
3. We hare «(,() == ((,()(l,() == 1. This reduces the compu
tation of «, {J) to the case where {J == 1 (A2 ).
S. COMPUTATION OF THE NORM RESIDUE SYMBOL
119
Before computing the symbol for the above cases, we make a definition. Let {J be a unit of k, {3 = ~:'=O a,,>." where a" are rational integers: Define D log {3 to be {J' /fJ where (3' is the derivative of the power serie:;, as discUS$ed in section 2. Then Dlog(3 is well defined mod )"t2 according to Lemma 2. Indeed, the different of k/Qt is )"t2 because k/Qc is tamely ramified. We have obviously Dlog(a(3)
== D log {3 + D log a (>/2).
Note that log.B may not be defined since {3 is not necessarily == 1 (A). Even in this log {3 may not be a unit, 50 D log fJ is not necessarily (log (3)'.
C8Be,
REMARK. The good radius of convergence for the log in ~ 2, or equivalently, if ord>. x> 1 then ordx >
Indeed, if ord>. x
k is precisely >.2. 1/(£  1) which is
the number of Theorem 1. We let S denote the trace from k to Qt. THEOREM 10. (a,{3) is determined
1.
0'
== 1 (A2)
(0', fJ) 2. a
== 1 (A).
=(is((log aD log p) •
Then
«(,0') 3.
according to the following scheme.
and {3 == 1 (A). Then
= (~S(Iog",).
0'== 1 (>.). Then (a,>.) = ,tS(t log,,>.
The previous statements hold in tlie sense that the exponents involving log a and D log (3 are well defined mod and hence ( can be raised to such exponents.
e
PROOF. We begin with the first formula and must prove that the exponent is well defined mod F}om a =: 1 (>.2) we get log a == 0 (A2) by Theorem 1. We know that Dlog{3 is well defined mod At2. Hence logaDlog{3 is well defined mod >.2.V2 = >.t = l>.e where e is a unit. We see that
e.
1
[8(£A7)
= 8(>'7) = 0
(mod l)
where "y is an integer. This proves that the exponent is well defined. From the functional equation of log and D log we see that the exponent is multiplicative in CI and (3, and that it is continuous in both arguments. It suffices therefore to verify our theorem for a multiplicative basis, i.e. for (f);, '7i) according to Lemma 3. In fact, since CI == 1 ().2) we may assume i ;;?: 2. Considering Theorem 9 we must prove that
E
j/r ==
~S«log'l1.Dlog'7j)
(mod l)
r.$~l
"+8j=£
We have
logf/i = log(l ).i) = _jAi 1 Dlogl1j = . IAi
"" /r LAn
.=1 ""
.
=  '" jA81 1 • L... .=1
120
XII. EXPLICIT RECIPROCITY LAWS
Multiplying the two series:
ls(ti+'il) = L ~. ~. S(0.'">+8j 1).
r,4~1
Let m
r1s~1
= ri + 8;. Let f!' I r but £,,+1 f r.
We shall prove that
_1_s().ml) == {I (l) tr+l 0 (t)
if m:::; t,
II =:
0
otherwise.
This will establish the fact that our two exponents are congruent mod £. We haw tv ~ r and 2ev ;,;; m because i ~ 2. Also, 2r ~ m = ri + sj. Case 1. 2 ~ m ... and 1/ = O. We expand Am  1 and take ( into the sum to
e
get
~s«>.ml) = ~s (~ (m; l)(_l)~(~+l) = r~ (m l)(_l)~S«I'+l). #=0
But m , t implies
1.10
+ 1 ::::: t.
1.10
We have
8«1'+1)
== {I
e 1
if J1. + 1 < t if J1. + 1 = £.
Indeed, if J1. + 1 < t then (1'+1 1= 1 is a primitive loth root of unity satisfying :1:1 1 + x f  2 + ... + 1 = 0 over Qt. Hence 8«1'+1) = _(coeff:l;l2) = 1. If 1.10 + 1 = then (I = 1 a.nd 8(1) = 1. We note that
e
e
0= (1_1)",1 =
'E (m 1'=0
Hence the sum is 0 except when m
!S«A t
m  1)
=
_!
'f (m  1)
tJo'=o
1.10
1) hence 141 ::: 1, because we have assumed
4. SPLlT'I'ING MODULES AND THE PRlNCII"AL IDEAL THEOREM
139
The abelian group B/IB is isomorphic to the factor wmmutator U/Uc of U under the correspondence
LEMMA 3.
group
GUIIUc ~
a+x II + lB.
PROOF. Define a map log: U ~ BIIB by loga'UII the log of the product is the sum of the logs. Indeed
== a+:c..
(mod 1B). Then
rogauab'U.. = log(abuall,,.'UII.. ) where88,
Zogaull +logbu.. == a+x", +6+x.. , and subtracting these two expressions we obtain an element of IB, namely
ub  b+ (XI1'l'
 X'"
+ a"" .. ) 
x" = (O'I)b + (0'  1)x...
Since B/IB is abelian, our log homomorphism induces a homomorphism log: U/U c  B/IB. On the other hand, we can define a homomorphism exp: B {
= 1.
U/UC bY putting
= aU e, aEA exp x .. = u ..U", 1';:' 1. expa
As usual, the second of these formulas holds for
u..
4
'7
= 1 as well becauae :1:,. = 0 and
This homomorphism exp Vll.l1ishes on I B, because exp(u 1)0 = auIUC = u".au;l(llUC =
and
exp(u 1)x.. = exp(xa.. 
X(7
uc
+ a.....  x .. }
== ~ ;; 1 (mod U"). 1£"1.1,,
Consequently, exp induces a homomorphism exJ): B / I B  U/ UC. A glance at the definitions shows that log and exp are mutually inverse maps; hence they are both
isomorphisms onto. This concludes the proof.
0
LEMMA 4. The transfer map VU,A: U/U c 4 A correspond.:!, under the isomorphism U/U" ~ B/IB, to the map S: B/IB  A which is induced. by the trace map S: B + B. (Notice that S carries B iuto A (cf. Lemma 1) and S vanishes ou lB.)
PROOF. For the representatives 1.1" we have
naa...
(viewing A in U)
=Ea. ..,.
(viewing A in B)
V(u,U C ) =
a
fT
= LO'x", 
.
¢...,. +¢..
:: LUX.,.:: Sx.,.:: Slog 1.1.. , t1
XUl. GROUP EXTENSIONS
and for a E A we have
V(aU C ) = Na (in U) =Sa (in B)
= i(a+ IB) = sloga. Since any element of U is of the form au." this concludes the proof.
o
THEOREM (Principal Ideal Theorem). Let U be a group whose commutator Then the transfer map Vu,u.: U IUc + U C I (UC)C is the zero map. IlUbgroup U C is of finite index and is finitely generote.d.
PROOF. Dividing out by (UC)C does not affect the transfer map, so it is no loss of generality to lI$ume (ucy = 1. i.e. UC abelian. Furthermore, replacing UC by an arbitrary abelillll subgroup A containing UC, we see that it is enough to prove:
THEOREM 7'. Let U be a group whose commtdator subgroup U C is of finite index and is finitely generated. Suppose that A is an abelian subgroup of U containing UC. Then, if e = (A : UC), e times the transfer map VU,A: U IUc + A is the zero map.
PROOF. SInce A :) UC,A is normal in U and the factor group G = UIA is a finite abelian group. Let its order be n (U : A). Then (U : UC) = (U : A)(A : UC) = ne. Let B be a splitting module for Ii 2cocycle coming from the extension UIA ~ G, as described in the paragraphs above. Then B is finitely genera.ted .over Z. because BIA R: I is a free Zmodule on (n I)generators, Alue a. finite group, and UC is assumed to be finitely generated. By Lemma 3, the factor group B IIBis isomorphic to UIU c , and 'is therefore a finite abelian group of order ne. Let hi, b2 I • • • I bm be elements of B which are representatives, mod I B, of a basis for BIIB. Let ej be the period of bi mod lB. Then ne = ele2 ... em. Let bm+l, bm+2,"" b. be generators for I B (as subgroup of a finitely generated abelian group, B,IB is finitely generated). Put f:m+l = em+2 e. ;;: 1. Then we have achleved the following three things: 1) The elements bl , ~,". lb. genera.te B.
=
= '" =
.:,
2) e,b; E IB, i
3) n:=l e;
= 1,2, ... ,8 •
= net
=
From 1) we have B ~j=l Zbjl hence IB that there exist elements OJ; e I such that
= ~j=llb;.
Therefore, by 2) we see
s
cib;
= L: 6;jb j,
i = 1,2, ... , •.
j=1
Putting "Yij :: e;a;i  Ojj, where O;j is the Kronecker delta, we obtain an 8 x • matrix (1';j) of elements of the group ring r of G such that B
(*)
E"Yi,ibj=O,
i=1,2, ... ".
j=l
Since G is an abelian group, the group ring r is commutative; hence the notion of a determinant of a square matrix with elements in r makes sense. If 6,;) is the matrix whose elements are the cofactors of the elements of the ma.trix ('Yii) we have
L ;YiJe"Y;j = 'Y • 5"i i
t
4. SPLITTING MODULES AND THE PRINCIPAL IDEAL THEOREM
141
where I = det "fi; and 0 ill the Kronecker delta. Multiplying (.) by ::rill: and summing
over i we obtain 'fbj =0 for all j. Since the bj generate 8 it follows that 'Y annihilates 8, and a fortiori, 'YB C A. Hence, by Lemma 2, I is a multiple of the trace, say"Y = tS. To determine the value of t, it is enough to compute the image of'Y under the ring homomorphism e: r ~ Z which is defined by GK, in which case K/F is a finite extension of degree [K: F] (GF: Gk), and AF c AK. The extension K/ F is normal if and only if GK is a normal subgroup of GF, and then its galois group GK / F , is isomorphic to the factor group GF/GK' In thi::; situation the finite galois group GKIF, operates on the multiplicative group AK of the normal extension Ki hence we have cohomology groups Hr(GK/F,AK)' It is this type of cohomology group in which we are Interested. However, if the fields K and F are finite algebraic number fields, the galois group GK/F operates not only on the multiplicative group of K, but also on the ideIe group of K, and on the idele class group of K i and in global class field theory it is essential to study the cohomology of all three situations. The ideles of the various finite algebraic number fields F can be assembled into one big group, the idete group of the field of all algebra.ic numbers, and the same goes for the idele cla..: G' R: G
I:
~A'
A
such that f({)'(l)a) = u'(fa), and such that), induces I). onetoone correspondence between the subgroups {G p'} and the subgroups {GF}' Then for each field F' we candefindF' by G p,,..,) = )'(G p,), and it is clear that !(A).FI) = ApI, G),K'/).F" = >'(GK'IF')' etc. Consequently, for each normal layer K' /F' there is an isomorphism
().,j).: W().K'/),F')
R:
H"{K'/F /)
induced by the isomorphism of pairs
I:
A)'K' ~ A~,.
If one formation is a field formation, then so is the other, and the isomorphisms (>', f). on the 2dimensional groups of the layers induce isomorphisms of the Brauer groups. If the formations are class formations, then one would also require that these isomorphisms between Brauer groups also preserve the invariants. When this is the case, it is clear that the two class formations are essentially the "same" algebraic structure and that any collBtruction carried out in one could be carried out in
XIV, ABSTRACT CLASS FIELD THEORY
ll;ti
the other with the "same" result. In particular, the isomorphisms constructed. in the main theorem would correspond, i.e. the following diagram would be commutative
Hq+2p.K' />.F')
Hq(G>'K'I>'F" Z) _ (>.,1).
t
(>',J).
t
H'l+2(K' /1"').
H9(Gk:'IF"Z) _
If (G,{GF},A) is any class formation, and .,. E G, then the "inner automorphism" .,1: (ir = 1'(11'1 _ r 1': a+Ta
ill an automorphism of it in the sense we have described, because by Proposition 5 of §3 we know that conjugation preserves invariants. Thus we have proved THEOREM 3. Let l' E G, and let KIF be a normal layer. Let 0: be the fundamental class of the layer KIF, Then r.o: is the fundamental class OfTK/TF and the following diagram is commutative:
Hq(GK/F,Z) ~ Hq+2(K/F)
~.!
~.
!
Hq(GrK/rP,Z) ~ HQ+2(rK/TF). The isomorphisms of the main theorem do not commute with inflation in p0sitive dimensions. The correct rul,e is given by THEOREM
4. Let F eKe L with KIF and LIF normal. Let oKIF and
OL/F be the canonical classes of the respective layers. Then the following diagram is commutative for q > 0:
Hq(GK/P.Z) ~Hq+2(K/F) (£:K]
inf!
Olt.,,, Hq (GLIF,Z) 
PROOF.
iUf! 2
Hq+ (LIF).
The theorem is an immediate consequence of the general formula
(inf 0:) U (inf {3) = inf(o: U {3). This rule can be proved by a dimension shifting induction, and it is also an immediate consequence of the well known formula for the cup product of standard cochains, in positive dimensions, namely
(f U g)(O'1o (12.· .. ,O'p+q)
= f(O'l' (12, .... (11') u «(71, •.•
I
(7p)g«(1p+l .... , (1p+q)'
Granting this rule, the proof is immediate; we have only to observe that the inflation of O:KIF is [L : KjO:L/F' Namely, these two classes have the same invariant, because 1
[K : FI
:=:
[L: K] [£: F) ,
and inflation preserves invariants. (It is the inclusion in the Bra.uer Group!) HellCe, for any (E Hq(GK/F,'l.) we have inf(O:K/F()
= (in!uK/F)(inf() = ([L: KjO:L/p)(inf(> =O:L/ p([L : K) inf (l
BXERCISE
1lI7
o
and this was our contention.
In dimensioDB r > 2, the inflation map in a class formation is very weak. For example we have COROLLARY. If F eKe L, and if the degree [K : F] ditlides the degree (L: Kj, then the inflationfwm K/F to L/F is the zero map in dimensions r > 2.
PROOF. Sillce the horizontal arrows of the commutative diagram of the preceding theorem are isomorphisms onto, we need only show that the left hand vertical map, [L : K) inf, is the zero map. But this follows from the fact that it is applied to the group HT(GK/F'Z), in which every element has an order dividing (G K / F ; 1) = [K: FJ. 0
Combining the downtoearth interpretations of the cohomology groups in low dimensions with the isomorphism of the main theorem we obtain the following speciaJ. results:
H4 (KIF) ;;:, H 2 (GK/F,'L)
r::.J
GKIF
H3(K/F} Rl Hl(GK/F'Z) =0 H2(K/F} ;;:, EfO(GK1F,Z} RlZ/nL HI(K/ F) ;;:, H1(GKIF,Z) = 0 (AF/NK1FAK) ;;:, HO(KIF) Rl H 2 (G K1F ,Z) ;;:, (GKIF/GK/F) «AK)NK/F/IAK);::: Hl{K/F) ;;:, H 3{G K/ F ,Z). By far the most important of these special cases is the next to the last. It is the socalled reciprocity law Uromorphism of class field theory, and the whole of the next section is devoted to a detailed study of its consequences. The following exercise, with which we close this section, concerns the case of H4(K/F). .
ExerciBe Using the isomorphism 0: Hl(Q/Z) r::.J H2(Z) we see that the elements of H'(KjF) are of the form a:. 5)(, where X E Hl(G K /F,Q/Z) and a: is the fundamental class of the layer KIF. If X(O") is the standard lcocycle representing X. then the map 0" + X(O") is a character of G, and we may identify X with this character as we have discussed. Thus the correspondence
X ..... a:. OX is an isomorphism between OK/F, the character group of the galois group of the layer KIF, and H4(K/F), the four dimensional cohomology group of the layer. a) Let FeE c K with Kj F normal. Show that the restriction from H4(KI F) to H4(KIE) corresponds to restricting the character X from GK / F , to the subgroup GK/E; and show that the transfer from H4(K/E) to H4(KIF) corresponds to the map of characters of the subgroup GKIE into characters of the big group GKIF, obtained by composing them by the group theoretical transfer. b) Notice that the restriction and transfer are weak maps in dimension 4; for example, if FeE c K, and the layer ElF contains the maximal abelian sublayer of KIF, then both maps are the zero map. (For the transfer statement, use the principal ideal theorem.)
tA
XlV. ABSTRACT CLASS FIELD THEORY
c) Let Fe K C L with K/F and LIF normBl. Show that the inflation map from H4(K/F) to H4(L/F) corresponds to the procedure of viewing a character of the factor group CK/F as So character of the big group CLIF, and raising it into the [L: K]tb power. (Use Theorem 4.) d) Given a standard 4cocycle f = !(O'l, (12,0'3, 0'4) of G K/ F in AK, representing a class (J E H4(K/F), show that the corresponding character X is given by
x(r) where /
*r
=invF(f * r),
is the standard 2cocycle defined by
(f *T)(O'l,C'2) ""
L /(0'1. C'2,p,r). p
5. The Reciprocity Law Isomorphism
In order that our nota.tion correspond as closely as pOBSible to that which will be used in applications, we will from now on write the formation module A mUltiplicatively. The effect of C' on a is then denoted by a". If ElF is an arbitrary layer, the corresponding norm homomorphism N 1::/ F: AE ..... AF is defined by Ne{Fa ilia"', where the elements C'; are representatives of the left cosets ·of GE in GF: GF = U O',G E • It is the multiplicative analog of the trace; if FeE c K and K/F is normal, then the map NS/F is what we have previously called the trace from the subgroup G K / E to the big group GK / P • For each normal layer K/F, the main theorem gives us a natural isomorphism
=
HO(KIF.)::.; H 2 (GK/P, Z). Both of these groups have aowntoearth interpretatioIlB. n,O(K/F) is isomorphic ro AF/NK/pAK. the factor group of elements in the ground level modulo norms from the top level; the isomorphism is induced by the map x: AF > HO(K/F) which is onro and has kernel NK1FA K . On the other hand, H2(G K / F ,'Z) is naturally isomorphic to the factor commutator group GK/FIG'k/F of the galois group of the layer KIF. The isomorphism is induced by the homomorphismJ (f > ( ... = 6 1 )K (a  1) which is onto and has kernel G'kIF. The special case of the main theorem mentioned above may therefore be interpreted as an isomorphism AFINK/FAK r:::.GK/P/G'k/F·
This is knoWll as the reciprocity law isomorphism, for historical reasons. Being an isomorphism between a factor group of AF and a factor group of GKIF. it is induced by a multivalued correspondence " .... CT, a E A p , (1 E GKIF between thol;e two groups. Then. by definition, we have
a +>
q
if and only if xa = a: . (cr,
where Q is the fundamental 2dimensional class of the layer K / F. A dual description is given by 3Here IS: H 2 (G,Z)
IG/ [~, IG/Fa.
We have Hl(G, T) :::: iIIomorpbism G/Oc::::
Z. Hl(O, I)Ls the connecting homomorpbiam of the exact sequence o .... fG .... Z[G)  Z + 0. and
88
Is well ktwwn, the map
(I .... ( 1 
1 (mod I)~ induces an
159
Ii. THE RECIPROCITY LAW ISOMORPHISM
PROPOSITION 6. Let a E AI" and u E GK / P ' Then a
invF(xa· ox)
(.)
t+
u, if and only
if
= xCu)
Jor all characters X o/GKIF. (On the left hand side of this equation, the chamcter X is to be interpreted as an element of HI (G, Q/Z) in the usual way, and oX is the corresponding element of H2(G, Z).) PROOF. Let Q be the fundamental class for the layer K/ F. If a ... u, then
invF(ka' OX)
= invF«(a. (a) . ox) = invF(a· «(a' cSx» = .!.O«a . X) = n
x(u).
This proves the proposition because the u's to which a corresponds are characterized by the values X(O') for variable X (00(: also pages 45). 0
Our next theorem concerns the commutativity of the reciprocity law isomorphism with various natural mappings between different layers. THEOREM 5. Using the symbol! (KIF) to denote the many valued COfTeSpO'II.dence inducing the reciprocity law isomorphism in the layer K/ F, we assert that the following dio.grams are commutative:
AI"
a)
inclU8ion
(KIF)l
AE !(KIEl
group theoretical
(GKIFIGK{F)
tr......f e r .
NEIP
b)
to
(FcEcK)
(OK/BlacK{B)
•
AI" ...E'' AE
t(KIB)
(KIF)!
GK / F c)
.. inclusion
(FCECK)
GK1E
AI"    ..'    .... AFT (KIF)! GK/F
d)
AF (L/FJ
Jt
GLIF
lCK?lrl c:onjug"tion by ..
•
identity
G
(FCK, TEG)
KT/F"
• AF !(K/F)
lI&tllral hom. onto ta AE induces the restriction from HO(K/F) to HOCK/E); while the norm map from AE to AF induces the transfer from (K I E) to (KIF). On the other hand, the restriction from H 2 (G K / F ,Z) to H~(GK/E'Z) is induced by the group theoretical transfer from G K / F to GK / E • while the cohomological transfer in the other direction is induced by the inclusion of GK/E in G K / F • Statement c) follows similarly from Theorem 3 of §4.
no
no
160
XIV. ABSTRACT CLASS FIELD THEORY
For d) we mUBt use another method because we have not introduced cohomology maps correspondi.ug to the natural map of AFINL/FAL onto Ap/NKjFAK, 8lId to the natural map of GLIF onto the factor group GK / P ' We use the duality criterion
of Proposition 6, which states that for a E Ap and
0'
E GLIF we have a I(LIF)~ u
if and ouly if
(*)
X(O')
= invp(xa. OX}
fur all characters X of CLIP. We must therefore show that if (*) holds for aJl X,
then (**) ?/J(O'G L / K ) = invp(Ka· o,p) for each character ,p of G K1P = CL/FIG L / K • To thi5 efi"ect, we let X
= inf01 A~F:B) J DF. In other words, in terms of the complements. the open sets A  N cover the closed set AB. In particular, they cover the compact set (A  B) n Up, where UF is the compact subgroup of AF mentioned in Axiom IIIe). Thus there is a finite set of norm subgroups N 1 , N 2 , ••• ,Nh such that the sets A  N; cover (A  B) n UFo The intersection N = Nl nN2 n .. · n Nh is a nonn subgroup such that (A  B) nUF n N is empty,
which proves that
n:=l
166
XIV. ABSTRACl' CLASS FIELD THEORY
i.e. Up nN c B. Now consider NnB. It is open and of finite index in Ap because both N and B are. Multiplying it by UF we obtain a subgroup (NnB)UF, which is open, of finite index, and contains UF. Such a subgroup is a norm subgroup by Axiom lIIe). Since the intersection of two norm subgroups is a norm subgroup, it follows that N n (UF(N n B)) is a norm subgroup. This last subgroup is easily seen to be contained in B if one remembers that we have constructed N so that N n UFeB. Thus B contains a norm subgroup and is therefore itself a DOrDl subgroup. This concludes the proof of Theorem 8. 0
CHAPTER XV
Weil Groups In this section we shall apply the abstract theory of group extern;ions developed in Ch. Xlll, §I, 2,3 to the case of a group extension belonging to the fundamental class QKIF E H2(KjF) of a normal layer K/F of our class formation. In doing so we will gain a new insight into the reciprocity law isomorphism. For the sake of efficiency a.nd ultimate clarity, our discussion will be quite formal. We first define a certain type of mathematical 5tructure called a Weil group of the normal layer KIF. We then prove the existence and essential uniqueness of such a structure. Finally we discuss various further properties of the structure. DEFINITION 1. Let Kj F be a normal layer in a class formation. A Weil group (U, g, {fE}) for the layer K j F consists of the following objects:
1) A group U (called the Weil group by abuse of language).
2) A homomorphism, g, of U onto the Galois group GK/F' Having 9 at our disposal, we can introduce for each intermediate field E between F and K the subgroup UE gl(GK/ E ). UE is the subgroup of U which is the inverse image, under 9, of the subgroup GK1E of G K / F • The final ingredient of the Weil group is:
=
3) A set of isomorphisms IE: AE ~ UEIU~ ofthe EJeveIAE onto the factor commutator group of UE, one for each intermediate field E.
In order to constitute a Weil group, these objects U.g. and following four properties: WI) For each intermediate layer E' j E. FeE is commutative:
c
E'
c
{/El must have the
K, the following diagram
UE/U~
!V£IIB UEIIU~,
where the left hand vertical arrow is the inclusion map between formation level. and the right hand vertical arrow VE'iE denotes the group theoretical transfer (Verlagerung) from UE to UE'. W2) Let u be an element of U and put fT = g(u) E GKIF' Then it is clear that for each intermediate field E we have U1; = UB'" Property W2) states that the 167
XV. WElL GROUPS
lCiS
following diagram is commutative:
where the left hand vertical arrow is the action of a on the formation level AB and the right hand vertical arrow is the map of the factor commutator groups induced by conjugation byu: UB > UUEU. l = UBa.
W3) Suppose £1 E is a normal intermediate level, FeE map g induces an isomorphism
c L c K.
Then the
UEIUL ~ GKIE/GKIL == GLIE which we do not bother to name. Since AL is isomorphic to UL/Uf by h, we can view UE/Uf as a group extension of AL by G LIE as follows: (1) + AL
(*)
~ ULlUf .... UBIU'j,
>
UBIUL
I':::
GLIE + (1).
The operation of G LIE on AL associated with this extension is the natural one, as one sees by applying property W2) to an element u E U1> having a prescribed image in G L / B (replacing the field E mentioned in W2) by our present field L). Property W3) requires that the 2dimensional class of our extension (*) is the fundamental class (XL/E of the la,yer L/ E. W4) We finally require that UK == 1. This concludes the definition of a Well group. Fortunately it is easier to prove the existence of Weil groups than it is to define them! THEOREM 1. Let KIF be a nonnal layer in a class formation. t.Xists a Weilgroup (U,g, {fEll for the layer KIF.
Then there
PROOF. Let U be a group extension of AK by GKIF belonging to the fundamental class (XKIF of the layer KIF (Cf. Ch. XIII, §1, especially Theorem 1). Thus, U is a group containing AK as llormal subgroup, together with a homomorphism g of U onto G K / F1 with kernel AK' Choosing for each u E GKIF a preimage tia E U such that (J' = g(u a ) we have then au" = uuau;;l = aU, for a E AK. and furthermore au,~ == U,,'lJ.rU;: is a fundamental standard 2cocycle of GKIF in AK' For each intermediate field E between F and K we put UE = gl(GKIE)' In the two extreme caBeS E = F and E = K we have UF U and UK AK. In general we have AK C UE C U with UEIAK 1'>:J G K / E , the isomorphism being induoed by g. The 2dimensional class of the group extension UE/A K I'::: G K/ E is the fundamental class (XKIE of the layer K/E, because it is the restriction to GK / E of the class OK/F, of our original extension, and we know that Q.j(/E = re8QKIF' Let us now consider the group theoretical transfer map VUE •AK , designating it by VK / E for short. We have discussed this map for the case of an arbitrary group extension in §2 of Ch. XIII. It is shown there that VK / E carries Ue/U'i; not only
=
into
AK
but into A~KIE
= AE'
=
We therefore view VK / B as a homomorphism with
xv. WEaL GROUPS
159
values in AE: VK/E: UE/U'E
+
A E.
In the corollary of Theorem 4, §2, Ch. XIII the kernel and cokemel of this homomorphism is analyzed in terms of kernels and cokerneis of the homomorphisms 03:
H 3 (GKIS,Z) + H 1 (G K1E ,AK)
and 02:
H 2 (G K1E ,Z) + no(GKIE,AK )
which are effected by cup product multiplication with the 2dimensional class of the group extension involved. In our present case this 2dimensional class is the fundamental cla5s of the layer K / E, and by the maiD theorem of class field theory we know that the maps Di r : Hr(GK1E,Z) .... Hr+2(KIE) are isomorphisms onto for all T. It follows that the transfer map is an isomorphism onto, VK / E : UE/U;, ~ AE' We finish the construction of our Well group by defining the isomorphism IE: As ~ UE/U;, to be the inverse of VK / E. All that remains is to verify that properties Wl)W4) are sa.tisfied, and this is not hard. Properties Wl) and W2) concern the commutativity of diagrams involving the isomorphisms /E. Replacing these isomorphisms by their inverses, VK/E, we see that WI) amounts to the transitivity of the transfer, namely
= VK/E'VEI/E(a). a E UE/U;" and W2) amounts to the rule, for q = g(u), (VK/E('»" = uVK/s(a)u 1 = VKIE'" (uau 1), aE UE/UB. VK/E(a)
which follows from the naturality of the transfer. 'lb verify property W3) we refer to Ch. XIII, §3, where the map v: H 2 (G K / E ,AK) + H2(GL/£,Ad
is defined. By the very definition of v we see that the class of the extension (*) mentioned in W3) is the image under v oftbe cl~s DiK/E of the extension Us/AK ~ GK/E. because the extension
UL/Ul .. UE/uf
+ GLIE
is a factor extension of the latter in the sense discUBsed in §3 of Cb. XIII, and the isomorphism ULIU'i ~ AL is given by the transfer VUL,AK' And since DiLlE = tJ0K/E it follows that uLIE is indeed the class of the extension (*). Finally we see that W 4) is satil:ified because by our construction UK AK and is abelian. This concludes the proof of Theorem 1. 0
=
Having defined the notion of Weil group, and shown the existence of Weil groups, it is natural to consider the question of isolnorphisms of Wei! group. It is clear how we should define isomorphisms, namely DEFlNlTION 2. Let KIF be a normal layer in a class formation. Let (U,g, {IE}) and (U',g'. U;".}) be two Weil groups for the layer K/F. Then a Weil isomorphism 'P from one to the other is an isomorphism 'P: U ~ UI with the follOWing two properties.
1'10
XV. WElL GROUPS
WI 1) The following is commutative
U""!'G K1F
1 rp
!
ideutily
"
U'_G K / F • WI 2) From WI 1) it is evident that cp(Ufj) = U~, for each intermedia.te field E between F and K, and consequently cp induces an isomorphism CPfj: UE/Ui ::::: UE/UEc. Property WI 2) requires the commutativity of
AE~UE/U'i;
l~ti~
!rpB
AE ...l!..... U'E/Uli lor eacl! E. THEOREM 2. There exi3ts a Weil isomorphism cp: U ~ U' for any two Weil groups U and U' of a layer KIF. Furthermore, cp is unique up to an inner automorphi5m of U' effected by an element of Ui.
PROOF. If cp is a Wei! ilKlmorphism then the following diagram is commutative, by WI 1) and by WI 2} for the extreme case E = K:
(1) 
AI{
!
{g
UK
~ U~ GKIF  (1)
(l)AK
1~.
1
Id.,·
rp
{s" Uk~UI~GK/F(l)
Conversely, we contend that any homomorphism cp: U ..... U' which makes this diagram commutative ill a. Weil isomorphism. Indeed, let cp be 5uch a homomorphism. Then, from the exactness of the rows, it foHows that cp is an isomorphism of U onto U', and by the commutativity of the right hand square we have CP(UE) = Ui; for each intermediate field E. Thus cp induces an isomorphism CPE: UB/Uj; ~ UB/UFf and we can consider the following cube:
AE
Sd. j
loci
a
AK
~
j"{"
UE/IUk
V
!id.• UjI{
AE , ~ AK
'!:
Y'E
Ui;/Ui;c
V'
,
'!.K ..
¥'K
Uk
The top and bottom faces are commutative by property WI) of Wei! groups. The back face is obviously commutative. The front face is commutative by the naturality of the transfer map V, because cP : U ~ U' is an isomorphism mapping UE on U and UK on Uk. The right hand face is commutative because the left hand square in the preceding diagram is commutative. Since the horizontal arrows are
e
xv. WElL
GROUPS
171
isomorphisms into, we conclude that the left hand face of our pube is commutative, and this shows that q; satisfies property WI 2) for all intermediate fields E and is therefore a Wei! isomorphism. • The problem of Wei! isomorphisms cp therefore boils down to the problem of middle arrows cp in the diagram at the beginning of this proof, that is, to the problem of homomorphisms between the two group exteIll5ions represented by the horizontal rows of that diagram. These two group extensions have the same 2dimensional class, namely OcK/F' Consequently, maps cp exist, by Theorem 2, §1 of Ch. XIII. Moreover, since the Idimensional cohomology group Hl(GK/F' AK) is trivia.i, it follows from the uniqueness part of that same theorem that I{J is determined uniquely up to an inner automorphism of U' effected by an element of U~ = f~ (AK ). This 0 concludes the proof of Theorem 2. THEOREM 3. Let FI C F e K e Kl with KdFl and KIF normal. Let (U,g, {IE}) be a Weil group for the big layer K 1/FI . Then (upfU'k,g, UdFCECK) is a Weil group for the small layer K / F, where 9 denotes the homomorphism of UF/U'k onto G K / F , which is indu.ced by 9 in the obvious way.
PROOF. This theorem is evident from the definition of WeibJ group. The lattice diagram at the right may help in visualizing the situation. 0
Now let KI/ FI be a fixed normal layer, and let (U,g, {fEl) be a Well group for it. In the next few paragraphs we suppose that all fields F, E, K, ..• under consideration are intermediate between K 1 and Fl' Clearly by choosing our fixed normal layer K)/ FI suitably large we can arrange that any prescribed finite set of fields F. E, K, ... are contained between FI and K I , and so are "under consideration". THEOREM
U~
"Uk
" (1)
4. Let E / F be an arbitrary layer. Then the following diagmm is
commutatwe: UF/U'f
.t (map
'.
induced by ) the inclusion UE C UF
UE/U,;; PROOF. This theorem follows from a certain simple propertyl of the group theoretical transfer when we analyze what it says. There is a minor technical difficulty arising from the £act that we do not aBSUnle E/ F normal. Because of this nonnormality, we must first choose a K (e.g. K = Kl) such that E C K and K/ F IThe generaJ property of trllonsfer which is essentiaJly proved bel~ is a$ follow&: If U ::) Ul ::) A are group6 with A IIobelia.n and normal of finite index in U. then, vieweing A l1li a. (UI A)module in the usual way. the tran5fer from V to A of aD element "I E VI is the nann from A.U,IA to AU/A of the transfer of "1 from UI to A.
172
XV. WElL CROUPS
is normal. We can then refer the things we are interested in to AK and UK IUK by meu.os of a cubic diagram of the following type:
AF
~ AK
Ina
t~
j '!:.
UF/U'}
V
A:=l~ '!:. i.
I
A:
~ UK/U'k
1
V
UE/U E
~
IN.
• UK/UK
Here we mlUlt explain Nl and N 2 • We choose Nl 50 tha.t the back face of the cube is commutative, going back to the definition of N EtF • Namely we write
GK/F
= UO"iGK/E
(disjoint union)
and put
Next, we choose N2 50 that the tight face of the cube is commutative. According to property W2) of Well groups, this can be done by choosing elements E UF such that g(v.)
v.
= CT, and defining
(mod UK) for
U
E UK (mod UK). Notice then that we have
UF
= UV,UE
(disjoint).
Now the top and bottom of the cube are commutative by property Wl) of Wei! groups. Since the horizontal arrows of the cube are isomorphisms into, the commutativity of the left side, which we want to show, will follow if we can show that the front face is commutative. This means that we must prove for u E UE that
VUy,UK(U)
= II Vi (VUS,UK (u»vil. i
To do tbis we write (disjoint). Then
UF = UViUE
= UViUKWj = UUK'lJiWj,
i
;'J
iJ
the last because UK is normal in UFo Now by the definition of the transfer we have
VUF,UK(U)
=
n
v.WjUWislV~l,
i ,;
where (i,j) ...... (i1.il) is the unique permutation of the pairs (i,j) such that each fa.ctor of this product lies in UK. But since UK is normal in UF we see that this
113
XV. WElL GROUPS
permutation is achieved by selecting first it so that il i. And now we are through because
=
WjUW;;.l
V
E K and then putting •
o COROLLARY.
IF and is induce isomorphism& AF/Ns/pAE :::;: UF/UBUP. N'E}F(l):::;: (UE nUF)/U~,
THEOREM 5. Let K / F be a nOTTnallayer. Then the reciprocity law isomorphism lor that layer is given by
. IF Ap/NKIFAK :::;: UFIUKU'j ~ GK/FIGK/p· where the right hand isomorphism is that induced by g. In other words, if
g.: UF/Up ..... GK/FIGK/F is the homomorphism induced bll g: UF + OK/F. then 9.1: AF + GK/FIGK/F is the norm re.sidue map. For each u E GK / F , select a representative u" E UF such that u = = iFl(uaUf,) E AF. We must then show that bO' corresponds to Wlder the reciprocity law map. From the commutativity of PROOF.
g(u,,), and let bl1 U
AF~UF/Uf,
!;ncl'K
/1 v AKUK Uk we see that
h(bO') == VIF(bO') "" V(Uu)
=
nuTUO'u;;Uk =iK(n a..,.,), l'
T
where we have defined elements Ur,,, E AK by a..,,, == IKl(uTtlau;~Uk)' Then a..,a is a 2cocycle belonging to the extension
AK ~ UF/UK.!!. GKIF and is therefore a fundamental 2cocycle for the layer Wei! groups. Consequently, from
II" =
KIF,
by property W 3) of
II aT,,, =Image of u under Nakayama map T
we can conclude that b" and u do correspond under the reciprocity law, as contended. 0
The theorem we have just proved shows that the entire theory of the reciprocity law is contained in the theory of Well groups. The reader will easily check that all the results of Ch. XIV, § 5, can be recovered immediately from our present theory. The reciprocity relationship between levels AF and galois groups G K / F becomes easy to visualize when one identifies AF with UF/U'F (by means of /F) and identifies GK/F with UF/UK (by means of 9). In this way, all the facts are wrapped up in
xv. WElL GROUPS
174
one neat nonabelian bundle, namely a suita.ble Wei! group U. From this point of. view we get one additional dividend. the Shafarevic Tbeorem.2 • THEOREM 6. Let F eKe L with K/F and L/F norma!, and L/K abelian. Then we may view G L / F as a group extension 0/ GLIK by G KIF so there is determined in a canonical way a 2dimensional class (3 E H2(GK/F,GL/K), the. class of this extension. By means of the reciprocity law isomorphism, GLIK ::::: AK/NL/KAL (which is a G K / F isomorphism), fJ determines then a class {3' E H2(G K/ F ,AK/NL/KAL). This class fJ' is the image of the fundamental class OK/I" E H2(G KIF,AK) under the natural projection of AK onto AK/NL/KA L. PROOF. The proof is evident from the lattice dia.gram at the left. We identify the various galois groups with factor groups of subgroups of U, and we identify AK with UK/U'K by means of Ix. Then NL/KAL is identified with UL/U'K, and the reciprocity law i50WOrphisw become3 the identity map of Ux/Uf, by the preceding theorem. Hence fJ' is the class of the extension
AK/NLIKAL
+
UF/UL
+
GKIF
and is therefore obviously the image of OXIF because oX/F is the class of the extension
UCK
Ax ..... UFIU'K
+
GX / F •
0
The theorem we have just proved shows that if K / F is a norma.l extension, and if B is a norm subgroup of Ax ·which is a. G KI ,,submodule, and if L is the class field over K belonging to B (so that B = NLIKAL), then we can determine the structure of the galois group CLIF in terms of objects associated with the layer KIF. Indeed, GLIF is i50morphic to the group extension of AxlB by G K / F belonging to the image of the fundamental class. In the preceding paragraphs we have seen how a Weil group for a big normal Jayer Kd Fl contains information about all intermediate Jayers ElF, and in particular contains, as factor groups of subgroups, the Wei} groups for all intermediate normal layers Kj F. This suggests that we try to go to the limit and construct one universal group, a Wei! group for the whole formation so to speak, which will have all the Weil groups of all finite normal layers as factor groups of subgroups of itself. This is the next step on our program. In order to carry it through we must assume that our formation is a topological formation (ef. Definition, eh. XlV, §6) which satisfies a certain compactness condition, a condition which is satisfied in local and global class field theory. We shall also view the galois group, G, of our formation as a topological group, the neighborhoods of 1 in G being the subgroups GF. Thus 2This theorem is due to ShafareYich. He observed that it is a ~uenoe of a simple relation between the AkizukiWitt map tI and the description of the Dorm residue correspondence via the Nal aO'l induces
180
XV. W£IL GROUPS
=
a representation of AdAK onto Ar:l, because A'K 1 1. Therefore, Ai: 1 is compact, being a continuous image of AdAK which is compact by hypothesis WT 2}. Consequently IGL/KAL is compact as contended. We have now shown that the family of topological groups {UK} together with the family of homomorphisms {lPL/K} satisfies Weil's conditions LPI, LPII, and LPIII". It follows that we can build a projective limit U with all the desirable properties one could wish for. We form the direct product
11 UK K
of our Weil groups UK, /Wd in it we consider the subgroup U consisting of aJl elements UKEU K
U=(11,K),
such that UK = IPL/KUL for all pairs L :) K. We topologize U by giving it the topology which is induced by the product topology in the direct product. At first sight, this means that a neighborhood of 1 in U is given by a finite set of fields K i , together with a neighborhood Wi of 1 in UK for each i, the corresponding neighborhood in U consisting then of the elements u = (UK) such that UK, E W. for each i. However, taking into account the "coherence" of the components of 11" i.e. the fact that UK, = IPL/K.UL for a suitable L ::) K; aJl i, we see that it suffices to consider the neighborhoods of 1 in U which are given by a single field L together with a neighborhood WI. of 1 in U L , these constituting a fWldamental syBtem. For each L we have a map IPL: U > UL defined by
!f1L(U)
= UL
for u = (UK) E U. Using the compactness of the kernels of the IDap5 'PM/L for fields M ::) L one shows easily that 'PL is onto, and consequently is a topological homomorphism of U onto UI.. We must now construct the representation g: U 0 G, and to do so we must use the hypothesis wr 3) which states that G is complete in the topology for which the subgroups GK are a fundamental system of neighborhoods of 1. This completeness assures us that G Is the projective limit of its factor groups GKI" = G/GK, 88 follows: Let 'l/;l./K! GLI"  0 GK/r. be the natural map. Then, given any family of elements {D"K}, with D"K E GK/k, such that UK :0 'lJ!L/KD"L for each pair L :J K, there exists a unique element GV can be factored as follows
up > G'fl/F ='H. 2(GK/p, Z) .!!... 'IfJ(G K/ P , DK) = (DK n AF}I DF !.!... G1 is surjective, there ,exists x E
that gab(x)
= rF(a).
uab such
Then
Vex)
  E Ker(TF);::: DF a
= NK/FDK. =
Hence Vex) =a.NK/Fd = aV(i(d» for some dE DK, and a V(~). Injectivit'll: Let u E U such that V(uUC ) = O. Then gab(uUC) TF(V(UUC» = O. Hence there is a d E DK such that U E i(d)Uc. because the k.eroel of 9 iu a Wdiagram is i(DK). We must show i(d) E UC. We know th~t 1 = V(uUC} = V(i(d)UC) = NK/Fd. By our hypotheses 011 DK, 'H,i(GK/F,D K } 0 and the cohomology of DK has period 2. Thus, 'H,l(GK/F,DK) 0 and consequently, Ii = rIv d~·l for some finite set of pajrs (d". all) E DK x GK / F . Let i(dv ) = II" and j(u,,} = q". Then i{d) = rIll UIIII"u;ly;! E UC as was to be shown. 0
=
=
=
That finishes our sketch of the existence and uniqueness of a Wei! groups in Weil's sense for layers in the special type of field formation we are considering, following Weil's proof for the fonnation of idele class groups of number fields. If, as in tbe case of number fields, the formation is a class forma.tion with cyclic layers over the ground field of arbitrary degree, then the class of the group extension of GKI F by AK given by the Wei! group, is the fundamental class. This was proved for number fields by Nakayama in a paper in the same Takagi memorial volume of the Journal of the Japanese Mathematical Society as [28J, by the same method we used to prove Theorem 8(b). The fundamental class was discovered at almost the same time by Nakayama and Weil, in completely different ways, Wei! as a byproduct of his discovery of the Wei! group ap.d Nakayama by a systematic study of the Galois cohomology of class field theory, partly in collaboration with G. Hochschild, leading to most of the cohomological results we have presented in Chapter XlV (d. [11]).
Bibliography III
.'
Y. Akizuki, Bme ~ Z~ tier Elemente deo galoil&chct Grvppe zu deft EIe
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