CLASS FIELD THEORY EMILARTIN
JOHN TATE
AMS CHELSEA PUBLISHING American Mathematical Society • Providence:, Rhode Island
2000 Mathematics Sullject Clas8i/ic4tion. Primary llR37; Secondary 11-01, llR34.
For additional information and updates 011 this book, visit www.ams.org/bookpages/cbel-366
Library of Congresa Cataloging-in-Publicatlon Data Artin, Emil, 1898-1962. Class field theory / Emil Artin, John Tate. Originally published: New York: W. A. Benjamin, 1967. Includes bibliographical references. ISBN 978-()"8218-4426·7 (a1k. paper) 1. Class field theory. I. Tate, JaM 'Ibm!nc:e, 1925- joint author.
n. Title.
QA247.A75 2008 512.7'4-dc22
2008042201
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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..
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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 p-extensions 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.ld-Wang Theorem 1. Interconnection between Local and Global m-th 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/chel-366/ 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, V-VIII, 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 Grunwald-Wang 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 1850-1950. 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; Non-abelian reciprocity laws and' the LUKlandll program; Iwasawa theory; Leopold's conjecture; Abelian (and non-abelian) t-adic repl'eIIeDtatiOnBj Lubin-Tate 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 Artin-Tate seminar on class field theory given at Princeton University in 1951-1952, 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 XIII-XV). 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, Springer-Verlag, 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 <:olllplcx tiold (:, a IInit,\! extension of the rational p-aA.lic field QJl' for some prime numhHr p, or " Ii.·lllllr rormal powcr series in one variable over a finite constant field. In other wordIi, II local field is a locally compact non discrete topological field. The idele group and the idele dass group oC Ii Klohtu fI ..1Ii k AI't'! dl1notP.l:I by J = Jk and G Ok respectively. The absolute vtUur. (or voluml') of lUI idille a is the product of the absolute values of its compOllentM: lal '" In view of the product formula, the absolute value of an idele c\l'prllllH ollly UII it~ c1B111l. We denote by JO (resp. GO) the group of ideles (re;p. illi~lt~ 1:111.'!."II'.11) uf IILriolutl! vluue 1. By a finite set 0/ primes S of II: we alwtLr.; IUCtLll 1\ 1I0'1'CIIIVty He't. r.ontaining the archimedean primes in case k is a. number field. Th
Q. or
=
n,la,I,.
Jk,S
= Js =
nk; n x
pES
IIp
p(S
consists of the ideles whose components are units for p ¢ classes,
Ck,s = Gs = k" Js/k·
1':::1
S.
The KToup of S-ide1e
Js/Ie' n J.q ~ J ..;jk;
consists of the idele classes which are representoo by S'-idClCl;. Notice that ki; = .k* n Js is the group of S-I.mits of Ie, i.e. elements of k whir·h 1U'f' 1IIIII.s for all II fI. S, and that GIGs ~ J / k* Js is isomorphic to thr. grouJl uf diviHe)r I'!IiIIIiCli in the Dedekind ring formed by the elements of k which are integral fCIT p 1/ S. We give J the unique topology BUch that, for each 8, J s ill npell ill J und the topology induced on Js is the product topology. Uniqutllle!ls bi'C8U&' J .:: Us Js, Existence because the group of units Up is open in kp for Men non'AI'(:hitnl'cll'an p. Note that J is locally compact because each Up is r.omlN\('t, IUld 1!AC:h Ie; ill l(x:lI.lIy compact. Applying the product formula to a-I for a E Ie, a .,. I, one PtmT.!'i that k is a. discrete subgroup of J. The theorems on fiuitenl'!.'<; of d_ nllmlwor and the existence of units caD be mJed to show that GO = J I) / Ie· iK ('()III ilIAd, or c()url'l!, C/eo R: J/J O is isomorphic to JR+ or to Z in the nnmoor field "lid fUII<:tion field cases respectively. If K is a finite extension of Ie, then there are injAr.t.ionll J ...... JK lUlU C" -+ OK, and in each case the smaller group map~ hOlllllomorphic,.)ly unLo /I, dOlied
2
PRBLIMINARlES
a
subgroup of the lArger one. If K/k is Galois, the Galois group operates on K, JK, and OK, and we have k R.: KG, Jir. R.: J~, and Ok R.: O~. The first of these isomorphisIrul follows from Galois theory, the second from definitions of ideles and of the imbedding Jk C JK, and the last follows from the first two in view of "Hilbert's Theory 90": [fl(G, K·) = O. If S is a finite set of primes p of Ie, we let the same symbol S stand also for the set of primes !ll of K which divide some prime pES. With tlili:! understanding we
have
= II K~
JK,s
x
IT U'J = II (II K~) II (IT U'J ) X
IPts
'JES
I>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 G-module "induced" by the Gcp-module 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 !p-factor. By the theory of local fields, we have Hr(G'P, U'P) ofor r > 0 if K<.p is unramified over k" and therefore if our set S contains all primes p of Ie which ramify ill K we have
=
W(G,h,s)
R}
IT W(G, II Kir) IT W(G~, K~). R.:
p~S
'PIp
PES
Passing to the direct limit over lw-ger and lArger S, we find for r W(G,JK) R.:!!mW(G,JK,s)
s
R}
> 0:
II W(G'P,K-!J) II
where U denotes direct sum, and for each p of Ie, !ll deuote5 a selected prime of K above p. In this wa.y, the Galois cohomology of the ideles Is reduced to the cohomology of the loca.! fields. The isomorphism is of course functorial and commutes with restriction, transfer, and inflation ill the k c L c K situatioll in the obvious way: A global restriction from GKlk to GK / L is reflected in the local restrictions from GK~/kp to GK~/L.' The global transfer in the other direction is reflected, for each p, ill the sum over the primes q of L dividing p, of the local transfers. The global inflation corresponds to the totality of all local inflations. Let n be an infinite extension of k. The idele group In is by definition the union of the idele groups JK of the finite subexteusions K/Ic of O/k, and the same goes for the idele class group Cn. If n is a Galois extension thell its Galois group a = GOlk
2. COHOMOLOGY
I
operates on In, On, and n°. For each K, Wi! hlt~ J« "~ Jg", whftrc G« - GnlK is the subgroup of Gn corresponding to the field K. ILIIII th" ill tniC for the idele class groups and for the multiplicative grOI1~ or till' lic!lctll. TIll"' for A = Jo. Cn, or 0', the mathematical :;ystem (G. {eK}. A) rtlllmlt'uLIi 11 "fllrllliltiou" in the sense of the beginning paragraphs of Chapter XIV. TIll' rf'lIoIlc'T miKht. wl'lI btlgin hill study of these notes by reading the first three ISc'C·tiolill of Vhl'plA!r XI V. whl'rt· the notion of class formation is defined in abstracto, bec:all/w' tht· lil'llL thrCJ(! r.haptcrs (Chapters V-VII) of these notes are devoted tCl the Imlllf Lhllt the IdCle dlL'lBe5 of global fields do constitute a c1&'l5 formation. Our notation and terminology for format.iOT1B is ra~her naive. For &II intents and purposes, the notion of a formation ill equivalcmt 1.0" pNr (G. A) c!OlIlIillting of Ii compact totally disconnected topological grolll' (; 1\1111 " C;'lIIncllllc· A 011 which G acts continuously in the sense that the stabili7.cr uf cwry 1·1f!lIItml. " l A is &n open subgroup of G. Given such a pair one defines cnhOlllolUItY KrollfJff H'·(G. A) for r ~ 0 either by using continuoU8 standard cochalllll, or cquivllll·lIlly. II.~ t.he direct limits under the inflation maps of the cohomology gl'nUjlfo //' (G/Il, A") of the finite quotients of G by its open normal SUbKreJU(lII (J. All fllnetun! of (G, A) these cohomology groups have all the properties of ordilllU'Y c:nhlllllll!uY.Y groups of groups, and the freedom to vaxy the "formation" moclult, A wd "(;'LI()iH group" G is a considerable technical advantage. For wore detaila the mader call conliuJt 121, Ch. Xl or for still more details, [22).
""'"l('
2. Cohomology In the proof of the main theorem, cohomology of finite groups.
p. 154,
we refer
to the
rollowi~ ""tilt
in the
THEOREM A. Let G be a finite group, and let A )( B - • C lie G G-pairing 0/ two G-module.s into a third. Let a E HP(G, A). Then lor each q E Z and I!(Jrl/ .•ubgroup S c G the cup product with the restriction of Q to S ~('Id., a IlUmvmu,p111sm Clq,S:
Hq(S,B) .... Hp+q(S.C).
Suppose lor some I/O that the maps Qq.-I.S are sufjective, the map' "",,,.'1 are bijective, and the maps Cl:qo+I.S are injective lor all subgroups S. Then the maps Cl:q,S CU'(; bijecnve lor all q and all S. The proof is not difficult. By dimension shifting, one reduces to the CIlIIC p = 0, in which case Cl:'l,S is just the map induced by 50Dle G·bmflilmorphilllll I: H - G. Adding to 0 a cohomologicaIly trivial module into whit:h B in,ll'('lll, ~ Kee that it is no loss of generality to assume that f is an injectioll. We theD coll8ider the cohomology sequence associated with the exact sequenr.e
0-1 B1.0 .... D .... O and see from our bypotheses that Hq,-I(S, D)
= H'lO(S, D) == 0
for all subgroups S of G. It follows now from the thc!ory of cohomolORir.IIl triviality that Hq(S, D) = 0 for all q and all S, whence the result. In discussing cohomologieal trivisJity one first redUl.'eK to Sylow groupe! "lid then, by induction in cyclic towers, to cyclic groups. where till' IIUlt.l.f!f III t.rivial. These methods have beeJl refined in recent workB of Nakayama alld lUw. The main
too
PRELrMINARIES
theorem itself is proved directly in Serre's Bourbaki seminBl report of February 1953 ([23], see also [21, Chap. IX]). At the time when Chapters V-XII of these notes were written, the isomorphism of the main theorem were not known except for q = -2, -1, 0, 1, 2, nor was it seen that these constituted PBlt of a gent'.ral pa.ttern. Indeed, the possibility of extending the cohomology of finite groups to negative dimensions was not known at that time. Therefore the all-important norm-residue isomorphisms (*)
GKIF/CCK1F
R:::
H- 2 {G KIF,Z) "'~~, HO{GK/F,AK) ~ AF/NK/FAK ,
(here G C denotes the commutator subgroup of a group G) for a normal layer K / F in a class formation was treated separately by itself in the missing Chapters I-IV. Two descriptions of it were given. The first method was to map u E G K1F onto the element fl.. aT,... E AF (mod NK/FAK), where {aT,a} is a fW1damental 2-cocycle for the layer KIF. Indeed, this Nakayama mapl is just an explicit formula for the cup product of the fundamental cla.ss with the class (" E B-2 (G, 'I.) corresponding to a, so that this method amounts to giving an explicit description of the map 0:-2 of the main theorem without recognizing it as a cup product. However in Chapters V-XII, the emphasis is placed on a dual description of the isomorphism (*), namely that of Propositions 6 and 6' of Chapter XIV, which involves no negative dimensional cohomology, even implicitly. Before explaining this method, we must introduce a formalism with characters which is also useful in other connections. Let G be a finite group. Since Q is infinitely and Wliquely divisible it has trivial cohomology, and the. exact sequence
0- Z- Q-Q/Z-+O gives an isomorphism
6: Hl(G,Q/Z) ..... H2(G,Z).
=
We can identify Bl(G,Q/Z) with the character group G Horo(G,Q/Z) and we then denote the image of a character X W1der the isomorphism 0 by aX E H2(G, Z). Thus if for each a E G we let x(a) be a rational number such that x(a) EO X(u) (mod 1), then 6:x(1', a) = X(1') + x(a) - X(ru) is a 2-cocycle with values in Z representing the class aX. Summing over r we obtain
L "X{1',a);: nx(a) rEG
where n = (G : 1) is the order of G. Thus passing to cohomology, it follo'Ml that the pairing
lThe IlIAP g ...... nrEG aT,a was introduced by Tadasi Nakayama in 1935 in his paper [191. He ehowed that if Kilt is a Galois exte1lSion with group G, and a ... ,T is a 2-cocycle of G in KO, then: (1) The map q .... TI..e G(1T... induces a homomorphism G - kO/NKo; (2) If the cohomology class of ar," has the maJ
=
a. is given by
~.
THE HBltBRAND QUOTlEN'I'
1("
U 0)( = nx(O') (mod nZ) ~ From tlilil formula we obtain a conceptual proof for the periodicity of the homology of cyclic groups:
.
1
c0-
THEOREM B. Let G be a finite cyclic group of ornrr n, a7ll1 A II G·module, and "" a generator of G. Let)( be the character of G 3urh that x("") ~ I/n (mod 1). Then th.e cup products with Ox and with ('I' give mutually Inlle:r,e ilomorphiSTn8 U6x
HP(G,A) ;::=:! HP+ 2 (C, A).
u<..
=
Indeed, we have (.p U oX = 6x U ('I' nx(",,) == 1 (mud n) In thbl cue. Now let K / F be a normal layer in a class formation I'M ('Ilc.h a ~ A,... we let (a, K/ F) denote the element of CX/F/GK/P corrtlllpontillljl; to the rcsidue class of a mod NX/FAK under the norm rf'.8idue isomorphi~m (.). Then (a,K/F) is characterized by the fact that
x«a, K/F))
= invp(xa UO)()
£Or ull XC GKIF
where xa denotes the ().djmensional cohomology c!MM rorrllllponc\1II11: to a. Indeed, if (a. K/ F) = 0' mod GC then by definition, xa "" aU'" whcrc u ill the fundamental cla.ss of the layer. hence
xa U OJ( = aU (" U OJ( "" nx(a)Q and this 2-dimensional class does have invariant )(0) lx.'Cauae Q bM invariant 1/n. 2 3. The Herbralld Quotitmt
The Herbrand quotient is used 50 frequenUy in dlWl fil'lC\ thl'lOl')' thAt WI! recall here its definition and properties. If f is an endoJIlurph"~1IL or IlII ILbl'\lllI1 Kl'oup A, we shall denote its kernel and image by AI and AI rC2t~~l.ivp.ly. . Let f and 9 be endomorphisms of an abelian group A lIucb ~IaAt /g • 0 .. gJ. Then the Herbrond quotient is defined by the expr!'SSion (A! ; AU) q(A) = q,.,(A):: (A, ; AI) provided both indices are finite. Special Case: G is a finite cyclic group of order n. A iIIll r..modu1e, / ... 1- cp ,and 9 = 1 + cp + ... + 'Pn - 1 where cp is & generator of G. Wel hnve
AJlA' ~ HO(C,A) ~ H'l(G. A)
A,/A! ~ H-1(G,A) ~ HI(CJ,A) and thUl:l the Herhrand quotient is denoted in thi~ C'IVW hy hl/ I (n, A) or hy hl/l (A), because it is the ratio of the orders of the 2- and l·diulcllIIiollAi t:UhUlIIUIt«y glOUpe, and so appears in many applications. 2The interpretatiou of the Nakayama map lUI the cup produl1 wlah ,_ &I ..... a quldc proof of the theorem of Akizuki-Nalcayama mentioned in thp prmri()\U Inet....l. danotlnll the class of /Ig, T by 0 and the cllISII of oX by 6)( we have
f,,,,,,,,,,.
=
((" . oJ· 0)( '"' (0' ( .. )·6)( o· (c.. ,6",). ,,~(,,) ... Hence. if 0 is of order .. = IGI and c;., . 0 = 0, then x(a) i. a" Inl...." I.•.• X(.) - o,b aIIl(. to tT E G'. (More generally. If Q ill of order m, then " .. 1m E G'.)
•
PRELIMINARIES
The basic properties of the Herbrand quotient result from the fact that it is
the "multiplicative" Euler-Poincare characteristic of the circular complex I
A~A 9
i.e. is the ratio of the orders of the two cohomo1ogy groups of the complex. Thus, by usual arguments, we find: THEOREM q.l. Suppose in the following diagmm the horizontal rows are emct onO the horizontal arrows commute with the vertical arrows:
0---- A' - - A ---- A"-- 0
I'!k ,ll, r!k'
0 - A' ---- A ---- A" ---- O. Then q(A) = q(A')q(A") in the sense that if any two of the three .quotient5 are defined then the third is also and the above equality holds. THEOREM
q.2. If A is finite, then q(A)
= 1.
[t is an elementary exercise to prove:
THEOREM q.3. If 9 and h are commuting end011Wrphisms of an abelian group
A, then qo,gh(A) = lJo,g(A)qo,h(A) in the sense that
if either side
is defined then the other is also, and equality holds.
We sketch 8. proof of the following theorem which generalizes a theorem of ChevaUey (Class Field Theory, Nagoya, 1953-54, Theorem 10.3). THEOREM q.4. Let G be a cyclic group of prime order p, and let A be a Gmodule such that qo.,,(A) is defined. Then qo,p(AG) and ~/1 (A) are defined and we have: ( J.._/ (G A»p-l 1Jo,,,(AG)P ''2 1
Let
I{)
,
=
qo,p(A) •
be a generator of G and consider the exact sequence
o~ AG =Al _¥' ~ A
I-¥'.
A1 -"" ~ O.
Since Al-"" is at the same time a quotient group and subgroup of A, we see that qo,p(A 1--",) is defined, because qo.p(A) is so by hypothesis. Since both of these are defined, so also is qa,p(AG) by Theorem q.l. But h 2/1 (AG) qo,,,(A G) because G operates trivially on AG. Hence
=
~/l(A)
= qo,p(AG)~/l(Al-rp)
qo,p(A)
= qO,p(AG)qo,p(A1-rp).
Substituting these equations in the statement of the theorem, we see that we are reduced to proving that h 2/ 1(A1-'I') is defined, and satisfies
~ (A1-'I')p-l 11
=
1
Qo,,,(Al-'I')
,
3. THE HER.BRAHD quoTIBNT
But the endomorphism] + rp + ... + 1(1'-1 annihilates AI -" 10 Ulal. we can view A 1-'1' as a module over the ring ZIXj/(l + X + ... + X P 1). I:e. we call treat rp as a primitive p-th root of unity, and we are to pl'OVl:
qo,p(A1-
(A 1_<;:>)),,-1. But this follows from Theorem q.3 because in the ring or integf'nl in thf' IInld of p-th roots of unity, the factorization of p is given by p = (1 - .,,)p-I t where E ill a unit. Another proof can be found in notes of Serre's COUI'lit' "HOlJloi
(k O : kO" ) in 8. local field k. Since this index is used at the end of ttM! proof IIf TRmma 2, Chapter VI, § 3 we outline the computation here. Let k be Il lIulI·ardlillmdean local field, let U be the group of units in k, and for each intr.gtlr , ~ I, let Vj . . 1 + p' be the group of units congruent to 1 mod pi. We let IIII' thl' "'''P f(z.) 1 for all Z E and 9(X) = ;t". Then
=
k.,
/lO,,.
(kO)' _ (kO : ''0") -
(k~: 1) .
We observe that the denominator is the order of the group in k. Now for any i,
flo,,,(k")
nr n th mots of unity
=Ilo,.. (k" IU)'lO,,,(U/ll.)flo ... (U,) :: qO, .. (Z)qO,n(finite
grOI1P)Ilo.,,(V,)
;;;: nqo,,,(Ui ). H i is large and n prime to the characteristic of k, thlm thr map z - zo.. mapa Ui isomorphically onto Ui+m where m ord~(n), as one IIfM'fI for IIxlUnple by the binomial series for (1 + t)" and (1 + t)l/". Hence
=
qO,,,(Ui) ;;;: (Ui : UHm) = (0 : p)m = (0 : no) where
Inlk denotes the normed a.bsolute value of n W :k
O " ) :::
I~~ (k~
in k.
1/1"1.
nllll!
finally:
: 1)
and in particular, if the n-th roots of unity are c.ont&lnlWi In ~, tlNm
(kO : k O") ::
I:'•. ~
These formulaa ca.o be checked directly in the art'.hinmdl!AI1 c''''', A: - • lUI" II: = C (recall tha.t in the latter case, the normed ab:;oII1t.I· valul' ill tI~ 1M11IQff. "' tIll' urdinary absolute value). They hold formall,. if the chlU'A("tnilltir of k dlvld.. n, bnc-AII8e
then
Inlt = 0 and (k"
: kO")
= 00.
•
PRELIMINARIES
4. Local Class Field Theory We now tWll to loca.l class field theory. Let k be a local field, and n its algebraic closure. We axe to show that the formation (Go/}" n·) is a class formation. If k is archimedean this is completely trivial, so we assume k non-archimedea.n. For any notmallayer KIF we have Hl(GK/F,K*) = 0 by Hilbert's Theorem 90, i.e. our formation is trivially a field formation in the terminology of Chapter XIV, because it is a formation of (multiplicative groups of) fields. Probably the best way to prove the Second Inequality
(F* : NK/FK*) t;; [K : F]
Ie the method of Chapter XI, because the detailed study of the norm mapping car~ ried out there is necessary for the ramification theory, and must be done sometime anyhow. On the other hand, there are short cuts if one wants only the inequality. As explained in Chapter XIV, § 3, one needs only establish our inequality for cyclic
layers of prime degree. Let KIF be cyclic of degree n. Then for any Bubmodule V of finite index in the group of units U of K, we have h 2 / 1(GK/F,W)
= h2/1(GKIF,Z)~fl(GK/F' UIV)h2 / 1 (GKIF, V) = nh2!1(G K / F , V)
and we shall prove this is equal to n by constructing in any normal layer KIF, cyclic or not, a subgroup V such that HT(G, V) = 0 for all r. Indeed, let {oa} be a normal basis for KIF. Replace 0 by 7r i (j where 7r is a prime element in F, and where i is sufficiently large so that if we put
M= ~OF~ ueG 7r0 K • Finally, put V ='1 + M. Then it is easy to that V is an open G-submodule of U, and moreover V is filtered by subgroups
we hil.Ve ]t..(Z C 7r M and M C !lee
~
= 1 + 'lfi M such that for each i, the module
~/V;+1 ~ M/lrM
is G-regular, and hence haJ3 vanishing cohomology. Now we can apply the following elementary lemma whose proof we leave to the reader.
Let A be a complete topological group and G a finite group operating A. Let A = Ao ::> 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 2-dimensional 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
H 2 (G K IP, K*)
- / G K F
:::::
1 -Z/Z. n
Composing these two we obtain an injection
invKIP: H 2(G K/ P ,KO) - Q/Z. For any a E F", and X E G, the class HO U ~x E 1/2((;K"" J(.) III repnwmtlld by the 2-cocycle aDx(O'.r). Taking ordinals, i.e. applyiD& tbe lDap XO - i Z, we get the 2-oocycle (ordK a)6X(u, T) which represents the cl~ ordK aU6X 6(xo,d,.. ") ill H 2(OK",Z). TIIIJII M have
=
(**)
invK/F(xa U OX) = Xo'd K "(!p) . X(1pord, .)
since ordF a = ordK a, our ex:tecsion being wlrluililicd. TbiI allUWI ,baL for unramified extension K/F, we will have
(G, K/F)
all
",II.·"
:: q;1\/~'
(In some classical texts, tbe oppOIIite sign ifl r.hOltlln.) F'lIrthMmnnl, IIII1t'l1'! OK'1" Is cyclic, every 2-dimemdnnal d"'~N i~ of the f(lrm Hfl U c\\ 1\1111 wr f'All 111M' t.lu~ rule (... ) to (ltjtablillh the requin,'(j prolJCrtiCli of iIlY,./". Jo'ilHtur all, lhft IIIVlU'ilUlt docs not change under inHation to a biggl'r unr8J11ifitlCI t!XlfOlllliun 1./10' wit.h L .) K :> 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 lIla-lITe. 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, bcc-AUHC (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 G-isomorphic 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 G-isomorphic 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<;e we wish to avoid the use of Haar measure, we give below a modified version of our preceding proof. We let k be a global field and S a finite non-empty set of primps (If k including all archimedean primes. The subgroups K', Js, KO Js of.J art' mvariant under G. (Recall that S also stands for the set of all the primes of K dividing those of k which are in S.) We have therefore
h2 / 1 (J/W) = ~/l(J!JsK·)h'/./l(J8K· /K"),
and it will be shown that both indices
00 the right are finite. From the finiteness of class number theorem we know that J/JsKo is a finite group, and consequently hZ/l (J/JsKO) 1. JsK* /K* is G-isomorphlc to JsjJs n K' Js/K.~. Consequently
=
h-J/l(J/KO)
=
= h2/l(Js)(h 2/ 1(K;W 1,
and it will be shown that both quotients on the right are finite. We can write
Js ==
(IT U~) x (11 K~) ~ES
q:lIlS
where each factor is invariant under G. By an argument llimila.r to that used in function fields, we have h2 / 1 (I1op~s Uop)
= 1.
We can decompose the finite product
IT K.p = n(IT 'Gr).
!pES
pes q:llp
Let K; be anyone of the fields K",. and let G, be its local grDUp. The aemilocaI theory
ShoWll
h2/1
that
(G, II ~eS
K~) = 11 h2/1 (0, pES
nK~) = IT
~Ip
"2/1 (G,. K;).
PES
From the local theory we know that h2 (G,. K;) = np is thf! InrAl df!gIa, and h1(Gp,K;) = 1 is trivial. Hence
ha/l(JS) =
nnp.
pES
We shall therefore have completed the proof of the finJt inequality if ... 1IKlCIeed in proving the following LEMMA. Let S be a fini.te 8et 0/ prime. 0/ k including all arrhimMMn primes and let K/k be a cyclic extension 0/ degree n unth group G. Tlv.n
~/l(Ks) = (11 np )/n pes
where n, (., the local degree [K" : k,).
14
V. THE FIRST FUNDAMENTAL INEQUALITY
PROOF. 1 Let s be the number of primes !p of the set S in K, and let JR' be the additive group of Euclidean 8-Space. Let {X'.Il}'.Iles be a basis of IRs and map the S-ideles of K into IR! as follows:
Cl -+ L(Cl)
=L
log IClI'.llX\ll. '.IlES This map is an algebraic homomorphism of Js into R'. We shall make it a Ghomomorphism by defining a suitable action of G on R". Namely we let X~ =X'U"
q E
G,
and extend q to R! by linearity_ Then G acts on IR", permutes the vectors X'.Il but JlOt necessarily transitively. We recall that la" l\}la = lal'.ll and using these facts we have L(u"} = log la"I'.IlXlP '.IlES
L
=
L log la"llPaX'.Il
o
lPes
= L log lal'.llX'.Il" '.IlES = L{a)". This proves that L is a G-homomorphism of Js into R'. The image L(Ks) is a lattice of dimension II - 1 in R', according to the Unit Theorem (see [16, p. 100J or [5, proof of theorem, p. 72J). This lattice is contained in the hyperplane R,-l of all elements £x
h:m(L(Ks» = h2/1(Ks) because "-2/1 of a finite group is 1. We now face the task of determining h 2/1(L(Ks»). We first extend the 8-1 dimensional lattice L(Ks) to an s-dimensionallattice M as follows. We let X = I:'.Il X'.Il' Then the vector X does not lie in the hyperplanelR8 - 1 spanned by L(Ks). We let M be the lattice generated by L(Ks) and by X, i.e. M:;;; L(Ks) + ZX
(Z are the integers).
Then M is a-dimensional, and spans R'. Since X" = X for all q E G, M is invariant under G and both L(Ks) and ZX are G-modules. The module ZX is
o
lThe key idea in this proof ill that for a fitrite cyclic group G and a finitely generated GJnodule M, the Herbrand quotient h2 / 1 (G, M) i. defined and depends only on the lR{GJ-module M ®l. R (_, for example, [5, Cb. IV, Sect. 8, Prop. 12j). In fact, if G is of order 11, genera.ted by v. and the characteristic polynomial of q acting on the vector space M ®1R is (x} = (x-1)"oI«x) with "'(1) '" O. then n" 1i:J(1 (G, M) = W{l)" For example, if 4>(x) = z" -1, then h:1./1 = n/r.
"
3. FIRST INEQUALITY IN GLOBAL FIELDS
G-isomorphic to Z with trivial action. Hence h2/l (ZX) =~. Furthermore, the above sum is c1elllly direct, and therefore ~/l(M) = h;z{1(L(Ks»' n.
The proof of our lemma will therefore be complete if we prove that
~/l(M)
= 11 np. ~es
For this purpose, we prove the following proposition. PROPOSITION. Given any s-dimensionallattice M in our R" that iI invariant under G, there eri3ts a su/}lattice oj finite index M which i3 InTJanant under G, and generated by basi3 elements Y!p (l:I) E S) on which the actlCJn vI G .., fIIfJen by ~
YJ =
y~..
(I
E G.
Before proving the proposition, we show how our IAmmA follows from it. Suppose we have found a sublattice M of M with the action de~(:rilll!d in the proposition. For each prime p, let Mp be the sublattice of it KI:III:raIA~1 hy /1.11 nlcments Y!p, \P I p. (In other words, break up Ai into transitivity dolUWI1lI UUUAr G.) Then each Mp is invariant under G and
..
M=EM, ill a direct sum. FUrthermore, e&::h pair (G,
Ai,> ill ..wlocllol.
By the eemllocaI
theory, we ha.ve for each
~/l(G,Mp>, = ~/1(Gqs,ZY,,)
Go,p acts trivially on the infinite cyclic group generated by Yo,p. COll8eQuently hz/ 1 (Mp) = np and ~/l(M) = Ilnp. Since if i:; of fiwl.e index 111 M, i~ foll~5
and
that
o
This proves our lemma.. PROOF OF PROPOSITION. For convenience
we define
IExo,pxo,p1 = max ~ o,p Ix., I
&
DOrm on R· by putting
z" £= R.
This norm is clearly invariant under the effect of q E G. Let b be a constant Buch that for any vector A (, R" there utll Y E M such that IA - Y! < b. Such a b exists since M is .9·dirnl'OIlLonru. For each prime PES let ji be one of the J.lcilUt:II \p I p. tAil Y, ( M be such
tba.t (1) Let Y., =
IY, -bX,1 < b.
1:pg=o,p .. eO Yf.
We contend that the
desired kind. We tint verify that
Wl.1.0TII
Y" poerate & IItlblattice of the
Ghall the proper efl',,'(:t.. Ltl. Y" ~ Y... Y,,;:: E Yt' == ,-",o,p
Indnod
u
V. TIm FIRST FUNDAMENTAL INEQUALITY
u was to be shown. All that remains to be done is to show that the vectors Y., are linearly independent. Given a relation
LX'IlY"=O
x.,
we shall prove that all = O. We note that the number of 0" E G such that p" = ql is exactly the local degree ",. Because of (1) there exist vectOrs Bp such that
Yp
= bXp +Bp
where IBp 1 < b. From the definition of Y., we get
L: bX; + jig='.P L~ = L: bX.,+C., jiO='.P
Y'.P =
jia=.,
=npbX'.P+C'.P 'where C'.P is a vector such that
(2)
0=
IC.,I < npb.
Substituting in the relation yields
L x'.PnpbX., + L x.,C1l. 'II
•
Let (C.,X'.P) be the component of C'.P along X'.l3' Since the vectors X\"jl are independent, the component of each X\"jl in (2) must be O. It is
X., [npb + {C1l,X'.P>] anil the inequality 1C1l1 < n"b shows that npb + (C". X.,) :/:., = 0, and the Y., are linearly independent.
cannot be O. Hence
This concludes the proof of the first inequality.
0
4. Consequences of the First Inequality We note that in Theorem 1, we have proved more than an inequality. Namely, we have actually proved that h2 (G, C) is equal to n· hl(G, C). This win be used at & later stage of the theory. However, at this point we shall only give applications of the inequality itself. Since G is cyclic, we have
h2 (G,C) = (CO: NO)
=(GoI:: NK//CCK)
or in terms of ideJes,
h2 (G,C)
= (J,,: k*NK/"h).
'Ib simplify the notation, we abbreviate NK/k by N, and locally write N'.P for
NKll / k ,. The first inequal1ty implies therefore that the nOrm index (J" : kN J K) of a cyclic extension is greater or equal to its degree. In particular, if this index ls I, then the cyclic extension is trivial, a.nd [K : kJ = 1 also. This remark is used to prove all important consequeIlce of the first inequality.
t. CONSEQUENCP,s OP THE FIRST JNeQUAI.ITY
1'1'
THEOREM 2. Let K be a normal extension 0/ a global field k. 1/ K '" k then there exi8ts infinitely many primes II of k which do not ..plit r:omple!ely in K. Phrased another way: If K'.j} = kp for all but a finite "um&t:r of primes P. then K=k.
PROOF. First let K/k be cyclic. Suppose K., shall prove that any idele a E Jk can be written 11
= o.N"K
11K E
= k, at almoet ail primes.
We
JK.
Let S be the set of primes where Kll '" kp • We eM find () E Ie suC'.h that 0.- 1" is so close to I at all PES that o.-lllp is a localnurm: U-1Qp = N<prt
0
in~
In the fol1owing special cases, one can use the Hilbert Theory to sharpen Theorem 2. THEOREM 3. Let K/k be II cyclic extenRion of primc poU/cr d"gre(! pro Then th.ere exist infinitely many primes 0/ k which do not ~plil 411 all '" K I i. e. which remain prime in K.
PROOF. K contains exactly one subfield For dl'!ll;lW'l" O\W almost all primes split in Borne way. Then for such prillll!tl p,
t.
Su~ that
fK , : kpl = [Kkp : kp) = [K: (K n k.. ») = r(KI~'i!f:kj is the local degree &I1diutrictlyless thanpr. Hence [(Knk,) : kj:> I andthf!refore Fe kp • This would mean that all but a finite Dwnlx:r of primes IIpl.it cumpletely in F, &I1d this is impossible. 0 The next theorem will be 1llied in the proof of the fIIlOOlid im!CluaiiLy. THEOREM 4. Let KI , ••. , Kr/k be r cyclic erlcf",iofU 0/ P""'I' d~ p which are mutually disjoint over k, i.e. K" n (KJ ... K" ... Kr > ... 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<• .. Ie, whirh is not the calle. 0
11
V. THE FIRST FUNDAMENTAL INEQUALITY
The previous theore1IlS concerning the exi5tence of primes splitting in a certaiD. way are weak instances of more general results concerning the statistical behavior of primes in normal extension, which can be proved using analysis. What we have proved in Theorems 2, 3, and 4 will suffice for our development of class field theory. We shall obtain one more consequence of the first inequality in function fields. THEOREM
field
5. Let k be a junction field in one variable
o1ler
a finite constant
100. Then there exist, a divisor 0/ degree 1 in k.
PROOF. We have already observed (Section 3, Theorem 8) that the degrees of the divisors of k form an ideal of the ordinary integers. This ideal is principal, generated by the positive integer 5, which is the greatest common divisor of all degrees of the divisors. It is also the g.c.d. of the degrees of the primes, and we shall now prove that 5 1. Let kd 100 be the extension of degrees 6 and let K = kkl be the corresponding constant field extension of K. For each prime p of k, the residue class field kp contains kl, and therefore for any \lJ I p, K'.Il = kl>' By the first inequality (Theorem 2) it follows that K k, i.e. 0 = 1. 0
=
=
It may be convenient for the reader to recapitulate the essential ingredients of the preceding proof: The first inequality in function fields is actually an immediate consequence of \he Riemann-Roch Theorem and of the fact that the constant field is finite. The only other facts used in its proof are the elementary properties of the index h2/ b and some elementary local properties. We see therefore that the existence.of II. divisor of degree 1 is fairly shallow. The existence of a prime of degree 1 is II. much more serioU8 question. Such primes do not always exist. and conditions under which they exist are obtained from the Riemann Hypothesis in function fields. We see that in function fields, the value group of the idele classes is the integers, and is generated by an idele class having volume exactly q. Denoting such an ideIe dass by at. so that lall = q, we see that we can write any ideIe class a uniquely in the form a = a.o where ao is an idele class of absolute value 1. We may therefore assign ordinals to idele classes, just as we did in the local theory, and we shall see later that from the point of view of class field theory, the group of idele classes behaves in exactly the same fashion as does the local group.
at .
·
I
.~.
"
CHAPTER VI
Second Fundamental Inequality 1. Statement and Consequences of the Inequality Let k be a global field, of characteristic p (== 0 or > 0). K/k denotes a normal extension of degree n with group G. We abbreviate NKI" by N and NK~/k, by
N'fl. The proof of the first inequality showed that if K/k is eydie, ~(CK) = (Ck : NCK ) = h1(CK)' n. In this chapter, we shall prove the SECOND FUNDAMENTAL INEQUALITY. For any normal exten.!Wn K/k the norm index (Ck : NCK) divides the degrre [K : kJ. Coupling the two statements together, we obtain inwlCdiately h 1 (CK) :: 1
for all cyclic layers of the formation of idele clasalB. In Chapter XIV, Section 2, Lemma 1 it is shown that the triviality of 'HI in all cyclic laye1'6 implies the triviality of 1il in all layers. Consequently we.get THEOREM 1. Let k be a global field and let K/k be a nomaal extension with group G. Then 'H,l(G, CK) is trillial. The formation 0/ icWe classes of the algebraic closure of k is a field formation.
If we consider the exact sequence
'Hl(JK/K*) ~ 1'f(K") ~ 1(2(hd we notice that the natural imbedding of 'H,2(K-) into 'H2(JK) mUllt hfl injective. We have already discussed the structure of 'H2 (JK) on page 2. It iH " weak direct product of the local groups 'H2(G p , K;). A cocycle dlWi (' c. 'H~(h) ill completely determined by its local components Cp. We obtain thf'Iefore THEOREM 2. Let K/k be a normal extension. Let c E 'H2(G, KO). Then c:: 1 if and only if Cp = 1 lor all primes p. In other words, a 2-cocllde 0/ K· splits in K i.f and only if it splits ellerywhere locally.
Let C E 'H2(JK)' According to the local class field tbIXlry, each local component ct> of c c&D be assigned canonically an invaria.nt Inv, (" We IIhall prove later (Reciprocity Law) that the cocycle clWiSe::i of K· are charlloC!tcri7M all tbullc cocycle cla8Se5 c E 'H2(JKl having Ep invp c = O. In order to prove the second inequality, we have to diHtilll(uiKh two cases depending on the characteristic of k. These will be 1I/1.lIclll'(l '1~'I"" I iVl!ly in Snctions 3 and 4. The Kummer Theory of abelian extension will IMl w"d ,n 11111 PTCMlf:. and is given in Section 2. In Section:> we point Ilut th"t tbe, 11I1i1l1~ divililbility of the group of universal norms in idCle cli\'~s(~s iN lUI imllll'liilltc (~I1"''(IIlCllee of the procedw-e used to prove the second inequality.
all
VI. SECOND FUNDAMENTAL INEQUALITY
We shall now perform certain trivial reduction steps showing that it suffices to # p, we may even assume
prove the inequality in cyclic fields of prime degree i. If i that k contains the i-th roots of unity.
LEMMA. If Elk is any finite extension of degree m, then (C" : NE/kCE) is finite and divides a power of m.
PROOF. To prove the norm index finite it suffices to prove it for normal extensions (because if K ::> 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 t-Sylow 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 i-contribution 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 n-th roots of unity. The n-th 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 l-cocycle 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) = ai-a 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 n-th root of uuity in k and consequently invariant under the action of G. Finally X(U)X(T) = X(O')(X(TW' = ol-a+"-,,.,. ~ 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 n-th power. The kernel of the ma.p 0 ..... 0" consists of the n-th 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 1-1 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 n-th 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 n-th 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 1-1 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 n-th roots of unit !I lie in Ie, p t n. There exists a 1-1 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 n-th 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<:p/kp is unramified if the value group of the valuation of K<:p is the same as that of kf>. Let Ii be one of the generators. Since Ii = (0 1/")" in K<:p, its value is an n-th power of a value. The same is therefore true in kp and this means that the ordinal of Ii is divisible by n : It = Tn. If 7r is an element of k with ordinal 1 at p then o(7r-")" is a unit at p and can be used as generator instead of o. 3. Assume conversely that .1../k*n has generators () which are units at p. It will euffice to show that each field k( W) is unramified. a = f/6 satisfies the equation f(x) = x" - {j = 0 and /,(a) = na,,-l is prime to p. By an elementary local criterion p is unramified. 0
b_ n = p, the characteristic of k. The elements of the prime field of k form an additive cyclic group of order n = p and we use it as value group for the characters of G. The theory goes over with very little change, one has only to be careful to replace the multiplicative group A by an additive one. This time a character satisfies X(CT'T) ::; X(u) + x(r) = x(u) + O'x(r) and is an additive l-cocycle of (G, K). The fi~t cohomology group in the additive setup is trivial again and one obtains X(O') = (1 - O')a for some a in K. This a is not arbitrary. To find the group A to which it belongs we introduce the additive operator p of K by defining: p(ft) = f31' -13· , If 13 is in the kernel of this operator then (3P - 13 ... 0 and this equation has as \;lution exactly the elements of the prime field. The operator p commutes with the elements of C. We apply now to the equation x(a) = (1 - 0')0 the operator p. The left side becomes 0 since X( q) is in the prime field and the right llidc shows now that p(o) E k. We define therefore A to be the a4dlhllc subgroup of those elements a E K for which pea) E k. If we form converaely with Rllr.h U1 element a E A the function X(q) = (q- l)a and apply p to it, then thl! right Hide becomes o and shows that X(q) is in the prime field (and couequel1tly left fixed by G). Its continuity together with
X(o') + x(r)
= XeO') + O'x(r) = (q -l)Q + «(77' -
v)a
= (a7' -1)a = X(U7')
ahows that this function is a character of G. The symbol (a,O') = (O'-I)a defines now a pairing of the IIdditi~ group A and the multiplicative group G into the additive group of til" prillII' IiI·lIl. The kernel of A is again k and we prove as before that the keruel of G is 1. We filld as before a 1-1 correspondence between the Bubfields of K Bnd the (additive) subgroups of A which contain k. The mapping of A into the ground field k is done by thl' opr.rnlor p. We obtain pA/pk'::!! and have a 1-1 correspondence bctwt.'(m tim ""btil!ldlf of Klk and the additive subgroups of B which contain pk given b,y U ... pit.
a,
VI. SECOND FUNDAMENTAL INEQUALITY
IT we start with an additive group A of k which contains pk, we form in the algebraic closure of k the group iA where the symbol ~(a:) means wl solutions of p(x) = xl> -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 1-1 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 n-th 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 n-th 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 n-th 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 81-unit), • 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 n-th 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 n-th power at all the primes of 82 and ai (for i '" 1) is an n-th power at. any qj I- IIi but not an n-th 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 n--th powers at each aj # Ilj and not an n-th 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 n-th power at every prime qi E 8 2 (i 'Ie 1). At qi every Uj with j 'Ie i is an n-th power and a; is not. This shows that 112,113,"" II" are divisible by n. Since n-th 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 n-th 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
= x-lODe _
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 n-th 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) "" =: 1I-1D(yz) = y-1Dy(z). This means in operators that D + x =
+ ¥z
,,-1 Dy and consequently for any n ;;.: 0 that (D + x)"
=:
y-l D"y.
Assume now in addition that 11 E P" and 11 f/. P,,-l. Then (D + x)"{I) = D"y(l) = y-1 D"(y) = 0 and similarly (D + x) ,,-1 (I) '" O. Assume conversely that (D + x)"(I) = 0 and (D + x)n-l{l) '" O. Put (D + x)"-I(I) = 1/-1.
,,-1
= ¥.
Then (D+ x)(y-1) = 0 = _y-2 D(y) +xy-1 and consequently x Therefore D +x y-l Dy, (D + x),,-l(l) =: y-1 DO-l(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),
k,
LEMMA
5. Let x be an integer of kp. Then 'Pp{x,y)
= 0 if and only if either
p lord y or x = pz for some z E kp •
k"
PROOF. Let x
Then 'Pp(x, e)
= 00 + alt + ...
with a, E kl' &lid 1/
=0 and consequently !pp(x,y) = ~ = nS(ao). p t
= t"t: where t: is a unit of
nix
=
'Pp Cx, 11) = 0 is therefore equivalent with pin or 8(00) 0 and we have to show that S(ao) = 0 is equivalent with x pz ;;1' - %. SUPP05e first x z1' - z. If the power series for % had poles then these poles would dominate in ;;11 and x could not be an integer of kp • If b is the constant term of z then ao = bP - b. Raising into the power p is the gcncratillg automorphism 0' of the finite field kdZ1' 80 that we have ao = (O'-1)b, an equation which is equivalent to S(ao) = 0 because the additive Galois cohomology groups of " field are trivial. If we have conversely S(aol = 0 then from the last rClIltLl'k ao (0' - l)b = pb for 8 certain b E kl. Let Xl = alt + a2t2 + ... and put ZI = -(XI + xf + + ... ). This series converges in the topology of kp and we get eaaily XI .zf - %1 = PZl and consequently
=
= =
=
=
xr
o p. A glollal pairing. The trace S in (1) can be hroken up into the trar.e Tp from the residue class field kl of kp to the constant field leo, followed by the trliCC S/ from ko down to the prime field 2:1" Let. V be the additive group of valuation vectors of k (see [2, XIII, 1)), { E V with local components ~p. We select one of the primes IIJld r.hOMe t l!eparating in k and local uniformizing parameter for this prime. Put )'C{) = LTp(reSp({p
dt».
p
),({) takes values in leo, is additive and ko-lin"M and jll not jcifmtically O. If I'plp is sufficiently small at all primes where t has polCli wld if (, is a unit at all other primes, then ),(e) 0, since each retliduc it! O. This shOWl! that A(~) is continuous
=
/'
ao
VI. SECOND FUNDAMENTAL INEQUALITY
on V. From [2, XVU, 31 it follows that >.({) vanishes for all { E k, i.e. >.({) is a d~~uuofk. Put now
j {dt = S'(>'(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 p-EXTENSIONS
=
We obtain for our ideIe: H~ == ~~. Put c = ,,-III; we find easily ~ _Z-2~1I+ O. Each component satisfies therefore ~ -= O. This means that each component i5 a p-th 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
,,-1
ft :::
=L
p
E
q)
q
where q ranges only over those primes where x has no pole and is consequently I!. local integer. Lemma 5 shows that each \Pq(x.lIaq) = 0, since x == zP - z. x is therefore in the kernel of k. 3. Suppose that x is in the kernel of k. Then 1'1(:1:. 0) 0 for all idcles, especially the local ideles ~ which have only one component :F 1. One obtain... therefore that tpp(x, ap) = 0 for all ap E kl" Let P be a prime where x has no pole. At such a prime Lemma 5 applies and tells that x = zp with zp E kp . Such a prime splits therefore completely in the global extension k(~x} and Theorem 2 of Chapter V Section 4 tells us that this global extension must collapse, i.e. x -= z1' - Z, Z E k. 0
=
z: -
"/. The Galois group of the maximal elementary p-exten.non. The p-extension with the Kummer group k shall be called the maximal elementa.ry p-extension. It is the field M = k{ik). The group A associated with it by the Kummer theory is A == ~k. This group A is paired with the Galois group 0 by (a, a) = (17 - l)c:;t. We ha.ve mapped A into k by the map Q -+ pa. The image of A is in our case the whole field k a.nd we ma.y view this pairing also as a pairing of Ie and 0 into Zp where
[x, 0'] = (17 -1) (~x),
x E k, C7 E t5.
The kernel of k is now pk, the same as in the pairing
Jr
(3)
1 p
O'(a); -z
-+
=
1 p
-x + \P(x, a).
The mapping (3) is actually a mapping of the idele c1a11t1C11lnw 0 with kernel We have therefore the theorem
cr.
THEOREM 6. The mapping (3) is II bicontmt.wu., ",amorphism between the fJf'O'IJfJ Jit/k . J[ (or Cltlcr) and the Calois group of the rna:nmal alwltan exten-
sion M of expomnt p.
VI. SECOND FUNDAMENTAL INEQUALITY
c. The proof of the second inequality. Let Klk be a cyclic extension of degree p. As subfield of M it is determined by an open subgroup ~ of 0 of index p. The norm group k .NK/k(JK) is open (all ideles in a sufficiently small neighborhood of 1 are norms) and of finite index in J k • According to the first inequality the index is at least p. The image of this group under the isomorphism of Theorem 6 is a certain open subgroup f)' of 0. The index of ~' in 0 is finite and at least p. ~' determines a certain subfield Elk of Mlk which is the compositum of cyclic fields of prime degree and e!J.Ch of these fields is left fixed by ~'. To each cyclic subfield K'ik of degree p which is different from K we shall exhibit an element of f)' which does not leave every element of K' fixed. This will show that ~' must be f) and prove thereby the second inequality since the index of f) is p. According to Chapter V, § 4, Theorem 4 there is a prime p of k which splits completely in K and remains prime in K'. If K' = k(~x) we can select p in lIuch a way that x does not have a pole at p. Let ap be an idele of k which haa at p a component of ordinal 1, and has components 1 at all other primes. It is a norm from K since p splits in K. a(ap) is therefore in Q'. Now we compute fP(x, ap) = cpp(x, ap)' p does not split in K', x is therefore not in pkp. ap has ordinal 1 (not divisible by p). According to Lemma 5 we have cp(x, Qp) '" O. This means according to (3) that u(a.,} moves ~x and is therefore not identity on K'.
5. Infinite Divisibility of the Universal Norms
It is 110 remarkable feature of the proofs of the second inequality that we obtain as a by-product the Existence Theorem, or rather the infinite divisibility of the universal norms of idele classes, which turn out to be the kemel of the norm residue symbol. We shall IlS8UUle here the norm indez is equal to the degree in all abelian extensions. This fact will, be proved ill the next chapter. (Up to now, it bas been proved only for cyclic extensions.) Under this assumption we shall prove THEOREM 7. Let k be a global field. Let a E Ck be an idele class which is a universal norm, i.e. a norm from every finite eztension. Then/or every integer n, CI ill an n-th power of a universal nonn, i.e. there exists a universal norm b E Cle .uck that a = btL.
PROOF. It was shown as 110 conooquence of the compactness of CO that the axiomatic conditions of Ch. 14 for the existence theorem are fulfilled. It suffices therefore to prove that for every prime n, a is the n-th power of an idele class. Furthermore, since we may prove this only in sufficiently large layers, we may assume that all necessary roots of unity lie in k. Let n be Ito prime. If n = p, the proff of the second inequality *c. above) shows that, under the isomorphism of Theorem 6, a universal norm corresponds to an element of the Galois group which leaves fixed every cyclic extension of degree p, and is therefore a Jrth power. We may therefore assume that n '" p. This puts us in the situation of Section 3. We go back to Lemma 2. We let 82 be the empty set, S ""' SI' Then Dl "" nPES k; x Dplls Up Js. Hence ~1 = ks and Kl = k(k~n). We also have Dl = k* Js = J. Since k* Dl = k* N 2 J2 it follows from the first inequality that K2 == k. The Kummer group of K2 is ~2 = ~ n k and heDce ~2 = k~.
"*
=
6. ANALYTIC PROOF OF THE SECOND INEQUALITY
.
Now from Lemma 2 we obtain (J : kD2)
33
.
." ..
= [Kl : kJ.
Supposing the first inequality to be true in all abelian extensions, we must have k" D2 = k" Nl Jl' (We have therefore explicitly constructed the clas:> field to k' D2' It is then no wonder that the existence theorem comes out as a by-product.) 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 n-th power at all primes in S. We shall prove that a is al80 an n-th 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 n-th power at all primes in S'. b differs from a by IIJl n-th power and therefore " is also n-th 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· s-1
pE
is bounded on 1 < IJ < 1 + E. The set of all primes has density 1. This follows from the fact tha.t the zeta-function (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 opr-n 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 L-functions, defined for Re(s) > 1 by L(II, X) = n,,~ (I - w.)1 have similar properties to those of Dirichlet L-functions. 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.
=
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 non-trivial 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.<; naturally the direct product R+ x U as an open subgroup with the product topology. It consists of those ideIes whose infinite component is positive and whose finite components are units. If a E Q is a given rational number then the coset Q(R+ x U) consists of the following ideles: their infinite component has the same sign 118 Q and each finite oomponent hall the same ordinal as Q. Since the ordinals and the sign of a rational number can be arbitrarily prescribed we see that JQ=Q~ xR+
xU
X V is open. Let k be any field. The exteIll>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 = t-1ut 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 mo-th 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 m-th 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(m-l 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 m-th 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 p-contributions to m of the prime divisors p of m and up the p-components 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
=
VII. RECIPROCITY LAW
The map u -+ 0'" is an algebraic homomorphism o([J into (5. Its kernel consists of those u which are == 1 modulo every integer m, i.e. u = 1. The map is therefore
an isomorphism into 15. It is continuous. Indeed a neighborhood of I in 15 can be taken to be the group ~". that leaves a given Q«() fixed and the inverse image of ~m is the set of all 1£ == 1 (mod m); this set is open in U. The image oHI is everywhere dense in 15 since every automorphism ofthe finite subfield Q(() is induced by some t1u' A.9 continuous image of the compact set U it is compact and therefore all of e5. Thus we have proved: PROPOSITION. The map u ..... au is a topological isomorphism.
alU onto e5.
By means of this description of the group 15 we define now a map« -+ 0'(0) of the idele group JQ onto 15:
=
DEFINITION 1. According to JQ Q x R+ x U we write a given idele a uniquely G = a . p. u where a E Q, p E 1I4 and u E fi. Put 0'(4) = 0'.. -1 "" 0';1. The action
on the roots of unity is therefore {,Cal ::; (,,-1. Since the map is a projection followed by an inver-lion followed by an exponentiation: a ..... u ...... 1£-1 ..... (,,-1, it is a continuous homomorphism of JQ onto 15 with kernel Q x R+. Since Q is in the kernel it is actually a continuous homomorphism of the idele classes CQ onto e5 with kernel DO = R+. (We note pa.renthetically that DQ is the connected component of CQ' Namely, R+ is obviously the connected component of JQ. The continuous image Dr), = 114 in the idhle classes is therefore connected and CQ/R+ is totally disconnected.) If ap denotes as usual the idele that has at p the same component as a and 1 a.t all other prime5 we may write for any finite set of primes
S is the complement of S in the set of all primes. AIl S becomes bigger and bigger
'a
approaches 1. Since a(a) is continuous, limo-(ca) "'" 1. A more highbrow and more formal argument is the following: The totality of all sets S forms Ii filter in the set of all primes. In this sense lims Cs "" 1 and the continuity implies lima(c s) "" 1. From this we deduce the convergence of the infinite produce illl a{ap) in the sense of this filter and the formula:
O'(a)
= 11 q{ap). l'
We compute now each single O'(ap): 1. p = Poo. Let the infinite component be eO: where e = ±1 and 0; positive real. According to our definition we have to split up \lp"" into three factors: (e, e, ... ) E Q, (a, 1,1, ... ) E 1R+, u = u- 1 = (I,e,e, ... ) E U. In order to find now (U- 1 we have to find an integer n == u- 1 (mod m) and notice that n e: will always do it. Therefore (U(Clp _ ) = (~ or:
=
a.
RECIPROCITY LAW OVER THE RATIONALS
39
2. p finite. We shall write idrues in this way: first the component of pog 8lId then that of p and then th05e of the other primes. Let .
ap == (l,pI'up , 1, 1, I, ... ). The three factors' are: (pI',pI',p",II', ... ) E Q (p-1',1, 1, 1, ... ) E lR+ 11,
= (I,Up,p-",p-", .•• ) E rt.
Since any root of unity ( is the product of an me·th root of unity (0 with P f me and Ii pr·th root of unity (pr it suffices to give the action of q(ap) on and CPr separately: We have to find an integer == 1£-1 (mod rno) and see that pP does it for all me if JJ ~ O. Hence (;(op) = The Frobenius substitution !{)p maps <0 onto (b and we may also write:
'0
(t.
;D(-..) _ '>0
,.'1':
-"0'
description which remains valid if JJ < 0 because of the group property. For the action on (1" we have to find an integer n:; 11,;1 (mod prJ i.e. simply a sufficiently good approximation of the p-adic number 11,;1 and we shall leave it at that. Then &
,....,(0,,) _
\p"
-
,._;1 _ t n ~p"
-
~r.
All this suggests the introduction of a. local symbol defined by: DEFINITlON 2. For every prime let up(ap) denote a map Op .... O'p(o,,) of into 6 defined as follows: 1. If p Pex" let ("oo(aool = ,sigD(a",,). 2. If p is finite and 0'" = p"rda. up, let the action on (0 resp. (1" be:
Q;
=
I'tTp ( 0<")
_ ,.v>p."d "'p
'>()
-'>0
I"a p ( ap)
''>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 p-component 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
=
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 mo-th root of unity (with p I'l'10) and (pr a pr-th 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(n-1pi') has the same action. 0
o
Q;
t
<
It would be quite nice if the next theorem. could be proved without appeal to the secolld inequality, but this seems to be hopeless at present.
3. I\EC'lPR.OCITY LAW
41
THEOREM 4. 1. Let K/Q be (], finite cyclotomic extensiqn determined by the BUbgroup ~ of 15. Then 0"(0) E ~ if and only if" E Q' N K/Q(Jx) 2. Let similarily Kp/Qp be a finite cyclotomic extension ofQp determined by the subgroup ij of 0 p. Again O"p(a,.) E ij if and 'only if a p E NK./Q, (K;). PROOF. Both "if" statements have been proved before. If M denotes therefore the ideles of Q (resp. elements of Q~) that map into ~ (resp. ~) then M :J Q-NK/Q(Jx) (resp. M:::> NK./Q;(K;». If we follow our maps by the canonical map of I!} resp. 0 p onto the factor groups 0/t) resp. I5p /G then M is simply the kernel. These last maps are onto the factor groups because the map O'(a) is onto, and O'p(op) at least onto an everywhere dense subgroup. This means that the index of M in JQ reap. Qp is the degree of K/Q resp. of KIl/Qp. By the global (resp. local) second inequality, the index of Q. NK/Q(Jk) (resp. NK./Qp(K;» is at most equal to these degrees. The theorem follows immediately. 0
The next and last theorem of this section is a purely local one. We have made several attempts to prove it by purely local means and have not succeeded. 1 THEOREM 5. The symbol O'p(ap } is equal to the local norm residue symbol (ap, Qp) in QJI' In view of the explicit character of the map O'p(a p ) (Definition 2) we have therefore before us an e;r;plicit determination oj the norm residue .symbol inQp • PROOF. For the infinite prime the statement follOWti from Definition 2. Assume therefore that p is finite. We select in Qp any element 71' of ordinal 1 (not necessar· ily p). We put O'p('/I') = u and ('/1', IQlp) =·T. We want to show that u = T. Let ~ be an open subgroup of 15, and KII the subfield determined by it. Both automorphisms 0' and T have the property that 1T is a norm from KII if and only if 0' resp. T are in ~. Denoting by ~(7 resp. QT the closed subgroups of 0 generated by these elements we see that ~ contains ~(7 if and only if it contains I}T' The two subgroups have therefore the same open neighborhoods and that implies 1)(7 = QT' ~(7 consists of all powers u· where r E Z;, the completion2 of the integers Z. It follows that T = (J'r. To prove that r = 1 we look at the action of 0" and T on an unramified field of degree m. Both u and T act like 'Pp (Frobenius substitution) since 7r has ordinal 1, ud T = O'r shows that 'Pp 'P~ on this field. Since 'Pp has period m on this field we get r == 1 (mod m) for all m. But this implies r 1. A unit Up is the quotient of ,.. = PUp and p (both of ordinal 1). We have therefore also up(Up} ,.,. (up, Qp) and the general statement is now an immediate
=
=
consequence.
0 3. Reciprocity Law
Let k be a global field. Let K/k be a normal extension with group G, degree n. Let p be a prime of k. The groups K'.jl arc isomorphic for all \l3 I p and so are the decomposition groups G'.jl. For each p select one of the primes \p I p lSince these notes first appeared there have been two local approachee fuund. The first was by Dwork [7], see ..Iso [21, Ch. XIll, Sect. 5\. The second was by Lubin and Tate [17). see also [12] aIId 15, Ch. VI, Sect. 3]. ~Completion with respect to the 8UbgrOUpe of finite index: Z Ii,!' z/.".z..
=
...
VII. RECIPROCITY LAW
and put K'.ll = K p , Gop = G p. 1{.2(G, JK) is a direct sum of semilocal components 1{.2 (G, Il~lp Kop) and each one of these is isomorphic to any of its local constitueDtS:
1{.2
(G, IT K'.JI) ~ 1{.2(Gp , Kp). '.JI!p
Hence 1{.2(G, JK) may be viewed as a. direct sum of the local cohomology groups 1i2 (G p ,Kp), by assigning to each cocycle class C E 1{.2(G,h) local components Cp E 1i2 (G p , Kp) which determine c completely. We have ~ = 1 for almost all p. We recall how these local components behave under inflation, restriction, and verlagenmg. 3
Inflation. Let L ::) K ::) k be normal over k. Let !P' I ~ be a prime of L. A local component of InfL C is the local inflation of a local component Cp to Lop,; (1)
(lnfe., c)p
=InfL". (Cp).
To discuBl'! the restriction and verlagerung, we suppose that H is a subgroup of G and that E is its fixed field. Let q I p in E.
Restriction. Let c E 1i2 (G,JK)' The local q component ofResEc is obtained by the local restriction of cp to Eq: (Resec)~
(2)
= Rese. Cp.
Verlagc1'lIng. Let c E 1{.2(H, JK ). A local p component of VE/kC is the product of the local verlagerungs of the components eq:
(VE/kC)P ~
(3)
n
(Ve./kpCti)·
~Ip
In the local class field theory we have attached to each element cp e 'H.2 (G p, Kp) a local invariant, inv" q,. These invariants are rational mod 1. We know that invp cp completely characterizes cpo In particula.r, Cp :::;; 1 if and only if inv, cp = O. We shall now assign invariants to global elements C E 1t,2(G,JK)' Namely we define iDvp C = inv" Cp and let
inyC =
L: invp c. p
This smn is finite because cp
= 1 at almost a.ll p, and hence inyP C = 0 at almost
a.lll'. We have trivially Tbema.p
c -+ invc is a homomorphism of 1f.2 (G, J K) into the rationals mod 1. We note that inv C does not characterize c and that iny C = 0 does not imply tha.t c = 1. (in contrast with the local !:let up). It is convenient to discuss immediately the formalism of these invariants, i.e. how they behave under inflation, restriction, and verlagerung. We note that the invariants we have assigned are functions of K and k. When necessary we shall write invK/k to emphasize their dependence on particular fields. 3 Verlczgerung is the term
uaed here for the map which today is called <:OrestriCtiOD.
3. RECIPROCITY LAW
The next three lemmas are immediate consequences of the local class field theory. LEMMA 1. Let L ~ K ~ k be nClrmal over k and let C E '}t2(G, JK). Then the invariant does not change under inflation to L, i.e. invp Infr. c = invp c, and
invL/k1nfLC= inVK/kCPROOF. From the local class field theory we know that the local invariant of cp does not change under local inflation to L,+," The lemma follows immediately ~m
0
REMARK. Lemma 1 shows that our assignment of invariants is consistent with the isomorphic imbedding of 1-£2(G, JK) in ,,-£2(15, Jfl), In other words, let Kdk and K2/k be two normal extensions with groups G1 and G2' Let Cl E 1t2 (G h JK1) and C2 E 1{2(G2, JK3)' Let L :: K t K2. If InfL Cl
= InfL C2
then invKl/" Cl :: invK2//;~' Hence invK/k is not a function of K but only of k. We may therefore write legitimately invk instead of invK/k' In particular, given c E 1{2(tJ, Jc,}. If ResK C 1 we know that c can be deflated to a cocycle class of 1{2{G,JK) (or equivalently that c is the inflation of a cocycle class of '}t2(G,JK»' To determine the invariant of a oocycle class c, it Buffices therefore to determine it in anyone of the layers on which it splits.
=
For the next two lemmas, we let H be a subgroup of G and E the fixed field. Let q I p in E and let nq = [E" : kp] be the'local degree of E. LEMMA
2. Let C E 1{2{6, JK). Then invq ResEc = nq illv, c, and
mv E ResE C= IE : kJ invk c. PROOF. From the local class field theory and (2), the statement concerning the local restriction is clear. We recall the elementary fact that E q1p nq [E: k] and therefore
=
~)nq illv.. c) = (2: n q) inv, C= IE: k) invp c. qlp
qlp
o
Summing over p gives the desired result. LEMMA
3. Let c E 1{2(H,JK)' Then the verlagerung preserves invariants, i.e.
invk VEI"c= invEC. PROOF. From the local class field theory, we know that the local verlagerung preserve5 the local invariant, i.e.
mvp V.G!
= inVq eq.
From (3) we obtain inv,(VE/kC),
= ~::>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 2-cocycle 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 p-component 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:x-x
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.
=
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.
=
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 en-I, 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, (7-1{ 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;d-l 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
=
map
c-+ mv"c Is therefore a. homomorphism of the regular elements R2(CK) into the rationals mod 1. In particular, if mv" c of; 0 then c of; 1.
3. RECIPROCITY LAW
49
LEMMA 4. Let H be a subgroup of G and let E be the 'fixed field. Let regular element oj'R. 2 (CK ). Then ResEc is regular, and
inY E ResE C =
c be
a
IE : kl inVI< C.
PROOF. Let L ::) K and suppose InfLc commutativity of Inf, Res, and j we have
= jc, with C E 'Ji2(G L,h).
By the
InfL ResE C = ResE InfL C = ResE jc = j Res E c. This means that ResE Ccan be assigned an invariant. By Lemma 2, we have inYE ResEc = IE: k] inYk C
and this proves our lemma, because we have already shown that the assignment of 0 invariants is independent of a.uxiliary fields and selected representatives. Our final task is now to prove that every cocycle class c E 'H,2(CK) is regular. At the same time we shall prove thai. the map
c-
invc
is an isomorphism: CK/k ~ !'lIZ, where n cyclic fields, and need a lemma.
= {K : k].
We shall first do this in
LEMMA 5. Let Klk be cyclic of degree n. There exists C E 'R,2(OK) having invariant lin.
Q
rtgUlar cocycle class
PaoOF. It suffices to show that if tin is a prime and I." I n exactly, then there exists a cocycle class c E 'Ji 2(JKl having invariant III.". Indeed. we let C = jc. Then inYl: C = 1I er by definition, and taking a suitable power product of such c for all primes t gives a cocycle class of invariant lin. Klk contairul a subfield Ko/k of degree f". According to the theorem on the ~teuce of primes, Y, Section 4, Theorem 3, there exists a prime p in k which remains prime in Ko: the local degree at p u, therefore f". According to the local c)a.<;s field theory, there exists a local cocyclc class Cp ha.ving invariant 1I er . We form the global cocycle claBs c having p-component cp , and 1 at all other primes. Then inv/; c = l/er, and the invariant of its inflation to K remains the same. This proves our lemma. 0 LEMMA 6.
Let K/k be cyclic of degree n. The map ;: 1i 2 (h)
-+ 1t2 (CK)
is unto. In other words, every cocycle class of OK is regular. Let C E 'H,2(CK)' Then inv" c = 0 if and only if c = 1. Hence 1t2 (C K ) is cyclic of degree n.
PROOF. According to Lemma 5 there exists a regular cocycle class c E 'R 2 (CK ) such that invk C = lin. The powers of c form a cyclic group of order at least n. Because of the second inequality and the fact that in the cyclic case, 1i2 (CK) ':!! C"INK/"CK, the group generated by C has order exactly n, and consists of all elements of 1i2 (CK ). If inYl:C" = 0 it follows that C" = 1. This proves our lemma. 0 We are now ready to generalize the above results to arbitrary normal extensions. Let K/k be normal of degree n. We shall prove that all elements of 1-t 2 (CK) are regular, and that 1i2(OK) is cyclic.
VII. RECIPROCITY LAW
Let K' be lilly cyclic field of degree n over k. (Tbe existence of K' follows from the proposition concerning the existence of cyclic cyclotomic fieldB.) Let L = K K' be the compositum. Let i! E 1i2(CKI) have invariant lin. Such a exists by Lemma 5. Then lnfL i! has also invariant lin. It follows from Lemma 4 that ResKlnfLc has invariant O. But LIK is cyclic. By Lemma 6 this implies that ResK InfL c = 1. This means that InfLc can be deflated to K, or in other words, that there exists C E 1i2(CK) such that
c
II1fL t = InfL C.
Since InfL c is a cocycle claBs of invariant lin, it follows that C is regular and has invariant lin. The group generated by c is cyclic of order at least n. The second inequality shows that the order is exactly n, and that if(CK) = 'Jf.2(CK) is cyclic. We have therefore shown that the map
c -. invk C is an isomorphism of }l2 (CK) into the rationals mod 1. If invite = 0 then e = 1. We now see that invariants can be assigned to every element C E }l2 (GK). The invariant is independent of the field L to which c may be inflated, a.nd this allows us to assign invariants to the elements of 1[2(16, Go). Consequently the map
c-+ illvl for C E '1{2(I!5,Gn) is an isomorphism onto the ra.tionals mod l. It is onto, because of Lemma. 5, and the existence of cyclic fields of any given degree n. The lemma on restriction (Lemma 4) holds in general. because'R2 '1{2. Hence we have proved the fundamental theorem.:
=
• THEOREM 7. Let k be a global field, and n its separoble algebruic closure, with group 6. Then (6, Gn) is a class formation. The element.'l C E 1['1.(6, Go) can be a8S'f/TI-ed invariants which satisfy Axiom 2 of Ch. XIV, 3 according to the procedure described above, and the map c -+ inv c is an isomorphism onto the rationals mod 1. THEOREM 8. Let k be a global field. Let K/k be a normal extension of degree n. 'Jt2(KO) is characterized as the sv.bgrOtlp of elements c E T{2 (JK) such that
invc = O. Furthermore, given local invariants AI' satisfying the conditions
1. ~Ap = 0 (np = local degree) 2. almost all Ap = 0 3.
I:p >'1> = 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<, and hence converges in~.
Since
(a,k)
=ll(Ilp,k,) p
on each finite subfield, this is also true on A." as contended.
o
62
VII. RECIPROCITY LAW
CoROLLARY 3.
The'nann residue symbol gives /I continuo14S mGP from C" into
6. PROOF. Let ~ be a neighborhood of 1 in e, i.e. an open subgroup of finite index. Let K be its fixed field. Let Cl E Jk. If P is unramified in K, and Q p is a. unit then (Up, Kp/k p ) = 1. If pis unramified, we have therefore (Up,k) C ~. 1£ p is ramified, it follows from the local theory that there exists a neighborhood N~ of 1 such that (Np, Kp/ kp) 1. Hence (Np, k) c ~. If we put N = TIp unr...,.. Up x TIl> 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 so-ca.\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 of1-t.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 1-1 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 con-esponds 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 non-abelian 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 non-abelian. 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 1-1 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 non-empty 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:
Jm-J",
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 m-th 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<.plp e<.pi<.p = deg(K/k) and hence deg(K/k) erpf<.p = deg(K<.p/k'Jl). If we now assume that Q E k;' then deg(K'Jl/kt.p) = 1, hence deg (K/k) = 1 or Q E km. 3. Assume (m E k. The field k( y'a) = K is then a well defined Kummer extension of k and our assumption about Q means that K'.JIm = kp for every p f/. S. Theorem 2 of Chapter V shows K = k hence a E km. 4. If (m f/. k we form the new ground field k' = k«m). The precediIlg part of the proof shows Q = fJ'" with f3 E k'. The polynomial x ffl - Q splits in k' into linear factors x - (j(t:.. Let xm - Q = i.(x) be its decomposition into irreducible factors in k; each J;(x) has a root of the form {3.; {3(!:.. and k(I3i)/k is abelian as subfield of k' /k. If a = 1;', '1, E kp,
=
n.
=
65
IX. CONNECTED COVPONENT OF IDtLE CLASSES
ni
then lib,) = 0 SO there exists one hex) such that li("1p) == O. This means that our p splits completely in one of the subfields k(13.) of k', Assume now that k' I k is cyclic of prime power degree. The subfields of ,., are then linearly ordered by inclusion and we may assume that k(PI) is the smallest of the subfields k({J;). H P f/. S then p splits in one of the k({J.) and consequently in k({J1). Theorem 2 of Chapter V shows k({Jt} = k, PI E k, a {Jr. 5. Let p be odd and put k1 = k«p). k'Ik1 is cyclic of prime power degree. pm with Considering kl as the new ground field we see that we can write a {3 E k1. Taking norms from kl to k on both sides we see that ad E k m where d is the degree of k1/k and therefore prime to m pr as divisor of p-l. Since am E k m we conclude Q E km. 6. Only the case m = 2t remains. The degree of k'lk is then a power of 2. If k' I k is cyclic our problem is settled by part 4 of our proof. This eliminates the case of a function field. We may therefore asswne that k is a number field, and that Jc1 Ik is not cyclic (especially that t > 2). Put k1 = k( J=1). Since it is well known tha.t k'Ik1 is cyclic we have kl '" k and we can write a f3r with PI E k i . Taking norms from k1 to k we obtain a 2 ::; {3"' with {3 E k. This mea.na a ±/32·- 1 and we must show that the - sign is impossible. Assume that it occurs. Then -1 (j3':-i' The assumption about Q together with t > 2 allows us to conclude that -] is a square at all primes p rf. S. According to Theo~ 2 of Chapter V we would have k1 = k which is not the
=
=
=
=
=
=
0
~.
REMARK. If the m-th roots of unity are not in k, the symbol k( 'V'£i) has no well defined meaning and a careless use of it may lead to mistakes. The crucial steps of our proof are those of part 4 and a reader that wishes to simplify them is asked to be especially careful at this place. In the next chapter we shall see by the way, that the conjecture that Q E k m is not a1wa.ys true.
=
We let k be a. nwnber field for the rest of this chapter. J Jk, G = Ok. The connected component of J is the product of the multiplicative groups of the complex local fields and the positive real lines at real primes. One knows from the elementary theory of topological groups that the connected component of G is the closure of the image of the connected component of J. However, we shall obtain a more explicit model for it, and will not make use of the abstract topological situation, which would in any case not be sufficient for a discussion of the cohomology of the connected component. The group U of unit ideles splits into a direct product fj x fj where fj Ilpao '"" .. and U = rrp~p"" Up. Accordingly each unit idel.e a can be written as a product CL "'" ii . Ii where has components 1 at all finite primes and ii has components 1 at all archimedean primes. The units e of the number field k, considered as ideles, can therefore be written accordingly e €e. The two factors on the right side are of course only ideles and not any longer principal ideles. The group U is compact and has a fundantental system of neighborhoods of 1 that are subgroups of finite index. It follows that the exponentiation with ordinary integers can be extended to an exponentiation with elements of Z where Z is the completion of Z Wider the topology of the ideals of Z. The symbol liZ is therefore
=
a
=
1. STRUCTURE OF THE CONNECTED COMPONENT
defined for every unit idele and every x E Z and especially ~.is defined for units e: of k and x E Z. It is of course aga.in only a unit idele of U. . As usuaJ we denote by rl the number of rea.! archimedean primes, by r2 the number of complex primes and put r :;: rl + r2 - 1. .' THEOREM
2. Let,,1o'" ,Cr be independent units of k. The com;sponding ideles Z. In other words, the
£1, ... ,er are independent relative to exponentiation with
relation
(1)
ZiEZ
implies that all x,
= O.
PROOF. The group generated (with ordinary integers as exponents) by the Ei is of finite index d in the group of all units of k. Let m be any integer. We find to each Xi in (1) an approximating ordinary integer v.: v. Xi (mod Urn), IIi E Z. Then e = e~r is an element of k and we can write
=
er ...
Dividing the right side by the left side of (1) we obtain
At a finite prime each €; has component 1; the remaining factors have exponents that are divisible by 2dm. e is therefore a 2dm-th power at every finite prime. Tjdm, " E k. This Tj must Theorem 1 shows that it is a dm-th power in k: c be a unit of k and coll5equently its d-th power in the subgroup generated by the e, : Tjd = e~1 " .c~', I1i E Z. Consequently c:;: e~le~2 . .. c~r = c~lme~·m •.. c~·m. Since Ci are independent, we must have IIi = l1im. This shows that every Xi is divisible by m. Since m was an arbitrary integer it follows that each = 0 as was to be shown. 0
=
x.
Let V be the integral valuation vectors of the rational numbers Q. We IJl8.)' view Vasa direct sum: V = Z + R (JR the reals) with the product topology. Any element>. E V can therefore be written as pair: >. = (x,s) where x E Z and s E R. The integers m E Z are canonically imbedded in V by the map m -+ (m, m). IT e is a totally positive unit of k (that means ep "" > 0 for every real Pool we define (not quite canonically) an exponentiation of e by an element>. = (x, s) of V by putting e.\ = €8 if" • The idele g8 is defined as follows: it has component 1 all finite primes and component c:.l"g~,oo at !'co; the log has any fixed value for a complex prime but is the real log if 1'00 is real (this laIlt restriction is necessary in order to obtain for the component an element of kp,.,). This c), is again an idele of U. It is the ordinary power if >. is an integer and has always volume = 1. Indeed le),l:;: WI = Ill' = 1. The map>. - 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 d-th 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,-Z1er-Z, ... 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,
,m
em
=
I::
=
lal'''+1 =
Let >.;
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 13-component, 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 G-isomorphic to the product II
'I1:- 1 (D/D')
->
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 <:p) let (Gp,4>,) 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 Grunwald-Wang Theorem l 1. Interconnection between Local and Global m-th 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 Grunwald-Wang 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 GRUNWALD-WANG 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 non-cyclic 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 non-cyclic. 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 non-cyclic 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.
Al-a is a 2t-th root of unity in k(i) = k(,,) and (,+1 f. k(i) (it generates F). Therefore Al-<1 = C:. In Q = A2' we can replace A by Al = Ac,;:. Then A~-a = (f+2>.. If II- were even we could achieve AI-a = 1 or Al E k contradicting Q ~ k 2'. So ~ is odd and now we can achieve that IJ is any given odd number especially m'. AI-a
= /'m' = ( 1 +(. ) ",' == (1 + r )(l-~)m'. '"
1 +(;1
It follows that {3 = A(t + (.)-m' satisfies {31-a we get Q = Qo{3'),' where Qo
(5)
".
=1 or f3 E k. Prom A =,19(1 +(.)""
= (1 + (,)m = 'I)~l "" (ifl.+1)'" = (2 + fI,)m/2 = (-2 - '1.)m I 2 = ({2 +fl.) 2' -) m' . 1
These various expressions follow from (3), (4) USing t ;. 8 ~ 2. This aD is in k om'; QQo 1 E k*m' together with QQa 1 = (32' shows QQ;l E k· ... or a E aok*m. Thus far we have seen that P(rn, S) C k m U exok.... We have still to investigate whether actually Qo Eo P(rn, S) and whether the two COllets are different.
1. INTERCONNECTION BETWEEN LOCAL AND GLOBAL ....TR POWERS
'rII
Let So be the (possibly empty) set of all primes p where F,/kp is of degree 4 (and consequently a four group field). Since unramified fields are cyclic, and k«.+t} is only ramified for primes dividing 2, and since archimedean extensions are of degree 2, it follows that So consists only of certain divisors of 2. Assume now that ao E k m or that it is in some k;' with pESo. Then it would be in k 2' resp. kf. Since m' is odd, (2 + 71.)2<-1 would be in k 2' resp. k~·. Extracting a 2t-l_th root (2 + '1).)( would be in k 2 resp. k~ where ( is a 2t - 1 _th root of unity of k resp. kp• But k(i) resp. kl'(i) are genuine extensions so <" can only be ±1. The fact that F/k resp. Fp/kp is a four group field shows that ±(2 + ".) is not a square and we have a contradiction. This shows that aokm 'i k m but it shows also that ao can be in P(m,S) only if So C S. Assume now So C S and let P s. Fp / kp collapses so one of the three quadratic fields of F must collapse. Either (8 or f/a+l or i'1a+l is in kp and a suitable expression from (5) shows ao E k;', ao E P(m,S). We have now proved the theorem:
rt
THEOREM
1. Let k be a global field, m an integer, S a finite set of primes 0/ all a E k· such that a E k;' for all p ¢ S. Then
GIld P(m, S) the group
P(m,S) = k·"" except under the follOwing conditions which will be referred to as the special ~: 1. k is a number field. 2. -1,2 + 'I. and -(2 + 71.) are non-squares in k, where s ~ 2 is an integer such that '1. E k, 1/.+1 rf. k. 3. m::: 2tm' where m' is odd, and t > s. 4. So C S, where So is the set of those primes p 12 where -I, 2 + '1., and -(2 +'1.) are non-squares in kp. In this special case P(m, S) = k'"'" Uaokom where 0:0 is given b1l (5). Obviously "aD E k*m/2, but 0:0 ¢ k O ",.
=
As an example let k = IQ!. Then 8 = 2, F = Q«(s) Q(i, -12). So contains the single prime 2. The special case occurs if 81 m and if S contains 2. '13 ::: -12 therefore
ao == (1 +;}m == (v'2)'" == (H)m
= 2im.
Taking especially m ::: 8 (the smallest integer where the exceptional case occurs) we have
ao
= 16 == (1 + i)8 == (v'2)8 == (-.1-2)8
and 16 is an 8-th power at 00 and at all odd primes. Indeed, if p is odd, the extenEion IQ!p(i, \1'2) is unrami.6.ed and must therefore be cyclic of degree 2. This means that either i or v'2 or A lies in Qp and the formulas show that 16 is an S-th power. On the other hand 16 is not an S-th power in Q or in the 2-adic field. CONSEQUENCE: Let K/Q be any cyclic field of degree 8. Then 16 is a norm at all p except possibly 2, and hence at 2 also by the product formula for the norm residue symbol. It follows that K2/Q2 cannot be unraznified of degree 8 (the ordinal of 16 is not divisible by 8). This shows that one cannot prescribe local conditions at certain primes and find a global cyclic field of given degree satisfying these conditions. The circumstances when this can be done are discussed in the next section.
x. THE GRUNWALD-WANG 'mEOREM AB a corollary to Theorem 1 we prove the following thl(Orem which gives us some insight into the structure of the idele class group: THEOREM 2. Let k be a global field, m an integer. Denote by Jm resp. C m the group of ideles resp. idele classes whose period divides m and let Jm be the canonirol image of J m in the tdele classes. We contend Cm = Jm unless we are in the special case with the additional restriction that So == IZJ (empty). Then there exists an idele class ao E 1m such that C m = Jm U aoJm. In function fields of characteristic p: C p = 1.
PROOF. Let a represent an a E Cm. Then am
= 13m
=Q
and consequently
0:
E
unless we are in the special. case with So :;:; IZJ when Q may be Qo{3m; this <:to is an m-th power at all p 60 there exists an idele no such that 0:0 = a(j'. The idele class aD of no is not in Jm or else no = {Joe with em = 1; this would yield 0:0 = /3lJ' which is not the case. We obtain now Om == 13m resp. Om = (aol1)m so that ~ resp . .8:0 is in Jm • This shows that a resp. ~ is in Jm . 0 P(m,IZJ). Therefore
0:
If k is a function field then trivially Jp = 1 and consequently Cp
=1p == 1-
REMARK 1. The structure of J m is pretty obvious: each component of and i.dele of Jm must be a local moth root of unity. Since the kernel of the canonical map of Jm onto 1m is very simple namely the moth roots of unity of the global field k (as far as they exist), the structure of Jm is also understood. Since So is not empty in case k == Q, this is not IW example for the special case of Theorem 2. Consider k == Q{ -17). Over ~ we have ..fi = iA where A E Q2 since -7 == 1 (mod 8). k«3) k( v'2, i) == F is a four group field which collapses over k2 = Q2v1 since i E k2 • Therefore So is empty, and we have the special case of Theorem 2 if we take m ::: 8.
=
REMARK 2. In the special case (if only t > 8), ao is independent of m. Fonnula (5) shows indeed that {ao)p = 1 + (. or 1/.+1 or i1/.+1 depending on which of the three quadratic fields collapses at p.
2. Abelian Fields with Given Local Behavior Consider the idele class group C of a field k. The corresponding local fields kp have been imbedded naturally in C. This imbedding is done in such a way that the local norm residue symbol (0:, kp) coincides with the global one (0:, k). An abelian exteI15iOll K/k is described by its norm group N NCK which is an open subgroup of finite index in C, and every open subgroup of finite index is conversely the norm group of exactly one abelian extension K/k. The norm groups NllK~ of the local completions are given by:2
=
N$lK~ "" N
n k;.
Let S be a finite set of primes. Via the injection P -- J" we see that C contains the direct product P = TIpES kp. If S contains more than one prime the topology that C induces on P is not any more the product topology. P will even not be closed in C. The intersection 2Thls follows from (VII, Sect. 3, Corollary I of Theorem 9).
2. ABELIAN FIELDS WITH GIVEN' LOCAL BEHAVIOR
defines a subgroup Po of P of finite index which is open in P. It describes the local extensions K!p / kp for PES and the interconnections between them. Indeed, N!pK~ = Po n kp and more generally, Po is the kernel of the norm residue ma.p for elements a E P:
(a. K/k)
= 11 (ap,K!p/kp) PES
Two questions arise now:
1. Let P ::: nPES k; but with the product topology. The map of P onto P is an algebraic i::;oroorphism. We may view P as a suogroup of J and the map as the canonical map into a. It is therefore continuous. Do the open subgroups of finite index of P and P correspond to each other? The answer is affirmative. 2. Let Po be a given open subgroup of finite index in P. Does there exist in open subgroup NCO of finite index such that Po = N n P? This would give the existence of an abelian extension K I k whose local behavior at the primes of 8 is the one described by Po. The answer is also affirmative. LEMMA 1. Let A ana B be subsw of a topological group, A compact and B closed. Then AB is closed.
PROOF. We have to show that the complement of AB is open. We remark that c I/:: AB is equivalent to A-lenB::: 0. We have therefore to show: If A-lenB ::: 0, then there exists a neighborhood V of 1 such that A -leV n B ::: 0. We denote by W's open neighborhoods of identity. Since A-I C is compact a change of notation traosfonns our qUestiOD into the following: Let A be compact and B closea and assume An B 0. Find a V such that AV n B = 0. Since B is closed we can find to each a E A a W~ such tha.t aW~nB 0 and a W.. such that WaWa C W~. Since A is compact we can cover A by a firrlte number ai Wj of the sets aW... Now we determine a V C 1\ Wi. If a E A is in ajWj then aV C aiW.V C a..WiWi C aiW~, and does not intersect B. 0
=
=
am
LEMMA 2. P n = pm unless we are in the special case with a nan-empty set So. In this special case let em be the idele class of P that has components ao at the primes of 80 and 1 at the other primes af 8. Then P n em = pm U empm.
PROOF. An element a E P n em can be represented by an idele It with components 1 outside S and we must have an equation of the form Il == atm. This shows that 0 E P(m, S) and therefore a === (jm unless we are in the special case when
=
that Il nm or, in the special case if So is not empty possibly = c".nm. Comparing components outside S we see that n can be taken in TIpEs k". Going back to P we obtain our lemma. 0 LEMMA
3. pm is closed in P and the /act()J' groupa PIP'" and
p/pm
are
compact.
PROOF. em is closed as we know. Therefore Pnem is closed in P. This shows that pm ]a closed in P unless we are in the special case. In the special case k is a. number field and the index of pm in P is finite. Since p2m C (P n a2m ) c pm and P n (!2m is closed, we see that pm is closed in this case also.
I
X. THE GRUNWALD-WANG THEOREM
The map of P/ pm. onto P/ pm is continuous. It suffices therefore to show that is I pm is compact. This amounts to show that kplk';' is compp.ct. For archimedean primes the factor group is finite of order 1 or 2. For non-archimedean primes kp ~ Z x Up, and therefore k~/k';' ~ Z/mZ x Up/UP', a product of a finite group and a continuous image Up/UP' of the compact group U". 0 THEOREM
3. The open subgroups of finite index oJ P and
01 P
correspond to
each other under the mapping. PROOF. Since the map of Ponto P is continuous, an open subgroup of P comes from an open subgroup of P. Let cOQversely Po be an open subgroup of fiWte index of P and select m ill such a way that pm C Po. Po/ pm is compact so the image Po / pm is compact and therefore closed. It is of finite index in P / pm and therefore 0 open. Consequently Po is open. LEMMA 4.
II Po is an open sv.bgroup
01 finite
index in P then PoC" is closed
inO.
P. Let W = fL,es Wp be a neighborhood of 1 in P Po, where the tV" are compact neighborhoods of 1 in kp. W is compact. Determine an m divisible by n such that pm C Po. Since Pol pm is compact one can find a finite covering Po = Ui p,FmW. Going over to P one PROOF.
Po
is open in
that is contained in
obtains Po = UPipmw. The point is that W is oompact as continuous image of a compact set (although not necessarily a neighborhood of 1 in P). Multiplying by en and observing that pm is absorbed by On. one has poen = UPiwen. W is compact and C n is closed. Lemma. 1 shows the contention. 0 • LEMMA
5.
o/en is compact.
PROOF. In number fields 0 has the structure R+ x 00 so that C/O" ~ Colce an image of the compact group 00. In function fields the structures are Z x 0 0 and Z/nz x Oo/Or; and the conclusion is the same. 0 LEMMA
6. 0 has a fundamental system
0/ neighborhoods
V such that the /Jets
onv are open subgro'Up~ 01 finite index in C. PROOF. Let W = np w. be a fundamental system of neighborhoods of J,
selected in such a way that the Wp are compact subgroups of k; if P is non· IIorchimedean. At the archimedean prime:> let them be small neighborhoods of 1. J"W is a group since for the archimedean primes p"Wp = k;n. The images Vof the W in 0 are a fundamental system of neighborhoods in 0 and the images of
J"Ware
onv.
Read modulo en the env are neighborhoods of 1 in 0/0". Since c/en is compact, a finite number of translations cover the whole group and this proves that the index is finite. 0 LEMMA 7. Let Po be an open subgrou.p 0/ finite index 01 P. There exists (il n is a given integer) an open sv.bgroup N of finite indez of 0 such that P n N Po(P n 0"). N contain!> 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 GRUNWALD-WANG 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 non-empty 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 non-archimedean 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 2-cocycle 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'WALD-WANO'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:: V-I :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 i-th 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(ua-a) = 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(#'Vt-l - #Vi) = #Vo + #Vl + ... + #\.'t-l - t#'Vt = =(#Vo -1) + (#V1 - 1) + (#V2 - 1) +. ,. + (#'Vt-l -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 k-l k* itself. We ha.ve
=
k-l :::> 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 - ITa-1 (mod q3) is Ii homomorphism of Yo into the multiplicative group of the residue class field, because rr u - 1 is a l-cocycle 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 p-group. 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 TOT-1CT- 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 $'1-2)
en + (cTI)i+l,Be-i
(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(g-i -1) (mod \)Ji+2). Replacing IT by (T-1q-11I) 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'T-1u-J
E
Vi+l.
This is the 0
2. If K/k is abelian, and Vi
PROOF. The commutator is 1. Let T be a genera.tor of Vo/V,. The period eo. By hypothesis we can select (]' i v,+1' This implies that eo I i, lIB CQlltended. 0
of
'T ~
H K/k ~ not abelian, we can generalize Corollary 2 lIB follows. Consider elements q E V. but 0' ¢ Vi+!. Let 'T generate VO/Vl . Note that r'q'T- V ¢ Vi+l for any integer &I, because Vi+1 is normal. Separate V. - V;+I into equivalence classes by letting a ... a' if a ' 'fvaT- v for some integer 11. Let the equivalence claBS of an element a bave or a elements. Then r u is obviously the leaBt integer /I such that a r"ar-". We have 'Trv(],T-rvO'-l 1, and since a iJ Vi+l we conclude from the theorem that .,.r.i E VI. Hence eo I iT" for each u. The above facts will be used in our discussion of the conductor later, to prove that the conductor is an integral ideal.
=
=
=
2. Ramification Groups of a Subfield
Let k be complete under a discrete valuation with perfect residue class field k, of characteristic p. Let K/k be a normal extension with group G. Let H be a subgroup, and E the fixed field. If E / k is normal, it will turn out tha.t the ramification groups of Elk are given by V;ll/H, but the nwnbering does not fit, i.e. V; is not necessarily ViH/H. We shall find the function of i which gives the precise correspondence between the l'amifica.tion groups. . LEMMA 1. Let DK "'" o[a], and DE
a(3 - {:J '"
= 01fJ]. Then for any a E G,
n (-ra - a) = n(0'1a - a).
'f'EuH
"/EH
PROOF. Let I(x) = Irr(a, E) = n-YEH(X - 'Ya). The coefficienw of f are in DE. We have J"(x) ~ Irr(aa,oB) = I1'1EH(X - a1a). But I(x) - /"(x) has coefficients divisible by a(3 - fJ and hence (a(3 - fJ) I I"(a) - j(Q) = /,,(a).
Conversely, let (3 = g(a), g(x) E o[xJ. Then 9 rT = g. Since g(x) - (3 has root a and coefficients in E, it follows that g(x) -fJ = f(x)h(x) with hex) E DdxJ. Hence g(x) - (3rT == r(x)hrT(x) a.nd g(a) - /3" = j'T(a)hU(a) = (3 -IJU. This proves our lemma. 0
.
Let 8 be a subset of G and let o(u, S) =
{I ES Then #8 = CT
o
aiS
.
I:rTEG 0(0', S).
Let i(a) + 1"" ordK(ua - a) for any number a giving a minimal bas~. We put 1(1) =
00.
Equivalently, we might have defined i(a) by CT E Vj(u) but U ~ V;(a)+l'
We obtain 00
(1)
s(a) + 1 == }:cI'(u, VII)' 11=0
2. RAMIFICATION GROUPS OF A SUBFIELD
87
Let Elk be normal, fixed under H. Let G = UHu. We have functions l(u) fur Elk. The following lemma gives l(u) in terms of the indices in K/ E.
LEMMA 2. If tlH n Va is not empty for some i, then _
1
l(u)
E (Vo: '" V'".. )
i(u)+1=
.. =0
'I#Jherr! 1(0') is the laryeat integt!1' v such that aH n V.. is not empty. We may also describe leu)
&!
the integer such that (1 E Vi(,,)H but (1 ;.
~( ..)+lH. PROOF.
H we take ordK (a(3 - (3) in Lemma I, we obtain
E (i(a"}') + 1)
e{i(u) + 1) =
-rEB
oc
:=
L E 6(a"Y,VII ) 1I=0-rEH l(")
. = L #«(1H n Vv ). 11=0
Suppose (1H n VII is not empty, say u'Yo E VII' (17 E VII
$>
Then
l7'YO"YOl"'( EVil
$> '"10 1"'(
E (O"'YO)-lv.,
$>"'(0]"'(
E
~
v.. nH
'.
"T E "'(o(V" nH).
This means that there is a 1-1 colTflspondence between elements of "To(V.. n H), (V" n H), and (aH n Vv ). Hence
-H(aHn VOl}
0=<
#(HnVv) =V".
Returning to our sum we substitute the number obtained and get 1(<»
e(i{u) + 1) ==
E #Vv • o
We note that e = # Va by definition. Dividing by e yields _
i(u) + 1 =
as desired.
l(u)
1
Lo (Yo: - V'".. ) a
XI. HIGHER RAMIFICATION THEORY
88
Let K/k be normal. Let x be a. real variable. If z is an ~teger ~ 0, we define the real fWlction CPKI"(x) by putting
CPKI"(x) + 1 =
:E'" (Vi 1: V. )' .,=0
0
"
We often omit the subscript of CPKjlr. and write briefly cp. We have
'"
cp(x)
1
= ~ (Vo : VII)'
=
We note that cp(O) O. We extend this fWlction by making it linear between integers. Its gra.ph looks roughly as follows:
tp(x)
lie
...... ~ .' . --~--~~--~--~------~-------z
i+1
Let {x}
= least integer ~ x.
Define Vo:
r
Vi=1
= V(:t}. Then it is easily seen that
1 tp(x) = 10 (Va: Ve) dt
x ~ O.
=
We' extend cp to the left by putting (VO : V",) (V", : VO)-l for x :::;;; O. Then tp(-I) = -1 and the slope of
Properties of cp(x). 1. cp(x) is continuous, strictly monotone increasing, and cp(O) = O. 2. cp(x) has right and left derivatives everywhere, cp~ and cpi. If x is not an integer, or if V", = V:c+l then
tp~(x) = cpi-(x) = (Vo ~ V.,)' At an integer i where \'; -::f \';+1 we have
= (VO ~ \';) and cp~(i) = (Vo :1\';+1)' cp'( -00) = f and cp'(+oo) = lie. tpt(i)
3.
4. cp(x) is convex. 5. cp(x) has an inverse function 1/I(x)
= l"-l(X).
2. RAMIFICATION GROUPS OF A SUBFIELD
89
Properties of'l/1(x). 1. rJo(X) is continuous, strictly monotone increa."!ing, o.nd ,p(O) = O. 2. rJo(x) h&i right and left derivatives everywhere, 1/1~ and 1/1~. By elementary calculus, 1/1'(x) =: 1/'(/("'(x», and 1/1~(x) or "'Hx) is always an integer for x ~ -1, except for I/1t(-I). 3. TfJ'( -00) = 1// and ,p'{+00) e. 4. TfJ(x) is convex. 5. If i is an integer, then ¢(i) is also an integer, for i ~ -1.
=
The properties of rp(x) and ¢(x) are obvious from their definitions, except possibly 5 which follows from 2.
",(x)
------;/f-----z
1// We may formulate Lemma 2 with the aid of the
I(o) + 1 == ,p(f(u)
(2) We use the notation W
I{)
function &i follows:
+ 1.
= I{)EIIc and cP = 'PK/E.
-.
THEOREM 6. Let K ::> E :::> k, with K/k and E/k both normal, E H. The romijic;q.tion grou.ps of E / k are given by
V.;;(.1:) == V:z;H/H,
or V",
.,"
fiwl under
= V;jj(:z;)H/H.
PROOF'. We have
a E V';;(:z;) 0# I(o-) ~ ip(x) 1,0(£(0'» ~ ip(x) £(0') ~ X 0# l(a) # {x} ~ 0' E V{z}H 0#
,
.
(by (2»
0#
o#aE ~H
thereby proving the theorem.
o
XI. HIGHER RAMIFICATION THEORY THEOREM
7. Let K
:J E :J k be as above. Then
= ip($(x» ;"(x) = ~(Iii(x». 'P(x)
PROOF. Both functions cp and ip($) are continuous, have value 0 at 0 and are differentiable except at a finite set of values. By the chain rule for differentiation, at the points where derivatives exist, we have
!
(ip($(x)))
= ~(~(x»W(x) 1
1
= (Vo : V.p{",}) (Vo : V",) 1
1
"" (VoH: V",H) (Yo
nH: V", n H)
1 - (Vo: V",)
= cp'(x). Hence at the points where the functions are differentiable, their derivatives are equal. By elementary calculus, it follows that cp = ip(~, as contended. In order to avoid mentioning explicitly the function cp, we may define V'"
= V",{z)
or
VII'("') ""
V",.
Then the result of Theorem 6 may be expressed as follows:
V"'=V"'H/H. The new numbering of the ramification groups is now set up in BUch a way that we may define ramification groups in infinite extensions. Indeed, let n/k be the algebraic closure of k (any other infinite Galois extension would also do). Let 6 be its Galois group, and let ~ range over normal open subgroups of finite index. For real x let !V~ be the subgroup of 6 containing ~ such that !V~/~ VZ for the nonnal extension b/k. Let ~o :::) ~. It follows from Theorem 6 that
=
n
!D~o
nb
= !V~ '1)0'
Define !V'" = !V~. The groupB for a finite normaJ. extension are now easily recovered from
!V%.
Indeed, we contend tha.t ~;o = !V'" . ~o. The inclusion :J is trivial. Conversely, let T E !Vfo' We can write T 'Tl)")'~ where 'Tl) E ~b and "Y~ E ~o· Hence 1"')';1 = T~ E !V~. The sets !V~ n '19~o are not empty a.nd are closed. Since the group is compact, the intersection n,c~o(!ln;n""lJo) c!V'" is not empty. Let cJ E !V'" be in it. Then cJ~o ",,1J0 for all I). This shows that T U-Y. -y E l)o, and proves our contention. 0
=
=
=
a.
The General Residue Class Field
The fields studied in this brief section will be used as the residue class fields of a compLete field in the next section. It turns out that local class field theory and the ramification theory can be carried out in these more general complete fields. We suppose therefore that k is a field satisfying Moriya's Axiom:
3. THE GENERAL RESIDUE CLASS FIELD
11
k is perfect, and for each positive integer n there exists exactly one exteuaioD of degree n of k. We note that any finite extension of k also satumes the axiom. LEMMA
3. All finite extensions of k are cyclic.
PROOF. It is obviously sufficient to prove that an arbitrary finite normal extension K/k is cyclic. Let G be its Galois group and let IK : kJ = n. For each divlsor d of n let 1/J(d) be the number of elements r1 E G with the exact period d. Any two a.utomorphisms with the same period must have the same fixed subfield. and must therefore generate the same group: they are powers of each other. Therefore "'(cf) ~ ip(el}. We have
E fII(d) =order of G = n = E ip(el}. din
It follows tha.t "'(d)
= ip(d).
din
Taking d = n shows that G is cyclic.
o
The following algebraic proposition will be useful in Lemma 4-
Let F be any field of characteristic p ~ O. Let £ be a prime f. p. Let a E F. The polynomial;r/ - a is either irreducible in F, or a is an i-th power in F. Let p > O. Let a, b, C E F antJ abc f. O. Let I(x) = azl' -In:. The polynomial f(x) - c is either iTTCducible in F or has a root in F. PROPOSlTION.
PROOF. Suppose 'z} - a is reduciple in F. The constant term of one of the factors with coefficients in F is of type (~ where r < t. and ( is an £-th root of unity. From ({/ii1 E F we deduce aT' E pt, 'frivially at E Fi and hence a(r,e) = a E Ft, as was to be shown. We turn to the additive case. Obviously, f(x+y) = I(x) + f(y). Let a be one root of I(x) = c. It is trivially verified that all the roots are given by a + 11(3 where 0,1, ... ,p -1 and (3:;:; p-{jbfa. Suppose f(x) - c is reducible. The coefficient of x1'- 1 in one of the factors in F is a sum of roots, and hence of type ra + 1J.(3 where 0 < r < p. It lies in F, and hence :(ro + 1J.(3) = a + (IJ.Jr){3 is in F and is a root of f(x) = c, as contended. 0
,,=
LEMMA 4.
Every element of k is norm from any finite eztt:n8ion.
PROOF. Using the transitivity of the norm and the fact that every finite eXtension of k also satisfies the axiom, we see that it suffices to prove the lemma. in case of cyclic extensions of prime degree t. Note that all £-th powers in k are norms. We distinguish three cases. t = p. Then kt = k because k is perfect. Everything is a norm. £ f. p and eis odd. Let a E k. Consider the equation xl - a. If it is irreducible, then a is clearly a norm. If it is reducible, then a is t-th power by the proposition, and hence a norm. f. p and t = 2. If x 2 - a is reducible then a is a square and hence a norm. Otherwise, its root generates the extension of degree 2, and -a is a norm. We shall now show that -1 is also a norm. Let K be the extension of degree 2 over k. We know that k has exactly one extension of degree 4, say K(..;a) with a E K. Let CT be the non-trivial automorphism of K/k. Then K(R) has al50 degree 4 over k
e
XI. HIGHER RAMIFICATION THEORY .
12
and hence K(-fo)
= K(.JQO).
By Kummer theory ai-a
Taking norms to k yields
= f32
fj E K.
=
1 (Nf3)2. Therefore N{3 = ±1. Suppose Nf3 = 1. Then f3::: "Y.,.-1 by Hilbert's Theorem, and substituting in the preceding formula gives (0"Y2)1-0' = 1 whence Q1'2 = a E k. We get K( JO) = K(.jii). But [k( Va) : kj ~ 2, and since there is only one extension of degree 2 over k, we must have .;a E K. This is a contradiction, and Nf3 = -1, as was to be shown. 0 LEMMA 5.
Ift:F p then (k : ~) ~ t.
1= 1.
There exists an element a f/: kt. The polyno1lli.al By Lemma 3, one root of this polynomial generates a cyclic field K of degree t which is therefore splitting field of the equation. This implies that a primitive l-th root of unity, lies in K. But (' satisfies an equation of degree < t over k. Hence' mU5t lie in k. We are therefore in the situation of Kummer theory, and since there is only one extension of degree (k : kl) t, as was to be shown. 0 PROOF. Suppose
(k : kl)
xl - a is irreducible by the proposition.
e,
=
LEMMA 6. Let ~ p.
a, bE k, ab 1= O. Let f(x)
= axP -
bx. Then the additive index
(k : f(k» is
PROOF. Suppose (k : f(k» :F 1. There exists an element C E k, C ~ f(k) and the polynomial !(x) - cis irreducjble by the proposition. By Lemma 3, one root generates a cyclic splitting field K of degree p. This implies tha.t f3 = p-V'bjO. lies in K, and hence in k, because f3 satisfies an equation of degree < p over k. It is easily verified that
p(,8x) = a-1flf(x) and we have trivially p(flk) = p(k). By the additive Kummer theory, and since there is only one extension of degree p, we obtain
(k: f(k»
= (k : a- 1(jf{k)) = (k : p(k}) ... p
o
as was to be shown. 4. General Local Class Field Theory
Let k be complete under a discrete valuation with residue class field k satisfying Moriya's Axiom as discW>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 n-th root. The final step is strongly ramified, and its group is a. p-group 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 t two monomials, this proves our lemma. 0
e-
Xl. HIGHER RAMIFICATION THEORY
IN
Let K/k be normal of degree I, and complete.ly ramified: p be the different. Then S(\ll~) = pr where r :=; [(n: + s)/lj.
LEMMA 9.
Let f)
= \lIm
PROOF.
= p~.
Let S(\J}')
= ~.
We must determine r. We have
p-"S(lPS )
= S(p-rlPs) = S(lP
8-
cD.
lr )
By the definition of the different, we see that \lIB-if' c f)-I and 1) C \lIlr-,. This me8llS that m ~ ir - 8, or equivalently, r ~ (m + s}/I.
Conversely, we have S(~)
rt pr+l => p-(r+1) S(l3') ¢. D ~
rt D rt 11- 1
S(1,l"-l(r+l»
~ ~-t(~+I}
rt
~ f)
~(r+l)-B
+ l)i - 8 > m + 1 > (m + 8)/1..
~ (r ~r
This yields r
o
= [(m + s)/t] as desired.
We are now in a poIIition to investigate the cyclic extensions of prime degree. Let K/k be cyclic of prime degree l, with group G. Since G has no subgroups except itself and 1, the pa.ttern of the ramification groups of K/k will be:
= Vo = ... = \It lit+l = lit+2 = ... = I,
G = V~1 where t is an integer
~
-1. The order of the different is
OrdK
iJ
=L
(#v. -1) = (t + 1)(£ -1),
i=O
according to Theorem 1. K/k is unramified if and only if t and unramifted cases are as follows:
:=;
-1. The graphs of ",(x) in the ramified
Unramified
Ramified
,p(x)
",(x)
slope t slope 1 -------~r-------~
-------,I'-----x
1) = (t,t) t/I(x)
= x for x ~-1
t/I(x)
= {:+I(X-t)
•. GENERAL LOCAL CLASS FIELD THEORY
THEOREM 8. Let K/k be cyclic oj prime degree
t. For an integer i
~ -1
VIe
have 1. NK"'(i) C k i and NK"'(i)+l C kt+l' I iIi I-t 2. (k; : k;+1 NK,,(i» ~ { f. .. il