[Grundlehren der mathematischen Wissenschaften] Algebraic Number Theory Volume 322 || Global Class...

Chapter VI

Global Class Field Theory

§ 1. Ideles and Idele Classes

The role held in local class field theory by the multiplicative group of the base field is taken in global class field theory by the idele class group. The notion of idele is a modification of the notion of ideal. It was introduced by the French mathematician CLAUDE CHEVAILEY (1909-1984) with a view to providing a suitable basis for the important local-to-global principle, i.e., for the principle which reduces problems concerning a number field K to analogous problems for the various completions Kp. CHEVALLEY used the term "ideal element", which was abbreviated as id. el.

An adele of K - this curious expression, which has the stress on the second syllable, is derived from the original term "additive idele" - is a family

a = (ap)

of elements ap E Kp where p runs through all primes of K, and ap is integral in Kp for almost all p. The adeles form a ring, which is denoted by

AK = IJKp• p

Addition and multiplication are defined componentwise. This kind of product is called the "restricted product" of the K p with respect to the subrings op ~ Kp.

The idele group of K is defined to be the unit group

h=AK· Thus an idele is a family

a = (ap)

of elements ap E K; where ap is a unit in the ring op of integers of K p, for almost all p. In analogy with AK, we write the idele group as the restricted product

J. Neukirch, Algebraic Number Theory© Springer-Verlag Berlin Heidelberg 1999

358 Chapter VI. Global Class Field Theory

with respect to the unit groups o~. For every finite set of primes S, I K

contains the subgroup

of S-ideles, where Up = K; for p infinite complex, and Up = lR~ for p infinite real. One clearly has

h = Ulk, S

if S varies over all finite sets of primes of K.

The inclusions K S;;; K p allow us to define the diagonal embedding

K* ----+ IK,

which associates to a E K* the idele a E h whose p-th component is the element a in Kp. We thus view K* as a subgroup of h and we call the elements of K * in I K principal ideles. The intersection

KS=K*n1k

consists of the numbers a E K* which are units at all primes p ~ S, p t 00,

and which are positive in K p = lR for all real infinite places p ~ S. They are called S-units. In particular, for the set Soo of infinite places, KSoo is the unit group oK of OK. We get the following generalization of Dirichlet's unit theorem.

(1.1) Proposition. If S contains all infinite places, then the homomorphism

A: K S ----+ fllR, A(a) = (log lalp)PES' PES

has kernel JL(K), and its image is a complete lattice in the (s -I)-dimensional trace-zero space H = { (xp) E flpES lR I LpES xp = o} , s = #S.

Proof: For the set Soo = {p I oo}, this is the claim of chap. I, (7.1) and (7.3). Let Sf = S " Soo, and let J(Sf) be the subgroup of JK generated by the prime ideals p E Sf. Associating to every a E K S the principal ideal ia = (a) E J(Sf), we obtain the commutative diagram

1 ----+ 0* K ) KS ) J (Sf)

1 A' lA 1 A"

o ----+ fl lR fllR i

fllR -----+ -----+ PESoo PES PESf

§ 1. Ideles and Idele Classes 359

with exact rows. The map ).." on the right is given by

)..II( TI pUp) = - TI vplogm(p) PESr PESr

(observe that lalp = m(p)-vp(a)), and maps J(Sr) isomorphic ally onto the complete lattice spanned by the vectors

ep = (0, ... ,0, 10gm(p),0, ... ,0),

for p E Sr. It follows that ker()..) = ker()..') = /.L(K), and we obtain the exact sequence

o -----+ im()..') -----+ im()") ~ im()..II),

where the groups on the left and on the right are lattices. This implies that the group in the middle is also a lattice. For if x E im()"), and U is a neighbourhood of i(x) which contains no other point of im()..II), then i- 1(U) contains the coset x + im()..'), and no other. It is discrete since im()..') is discrete.

For every p E Sf, if h is the class number of K, then ph belongs to i (K s), i.e.,

J (Sf)h ~ i (K S ) ~ J (Sf),

The groups on the left and on the right have rank #Sr, hence so does i(K s). In the sequence (*), the image of i therefore has rank #Sr, and the kernel has rank #Soo -1. Hence im()..) is a lattice of rank #Soo -1 +#Sr = #S -1. It lies in the (#S - 1) -dimensional trace-zero space H, since TIpES I a I p = TIp I a I p = 1

for a E K S• 0

(1.2) Definition. The elements of the subgroup K * of I K are called principal ideles and the quotient group

CK = h/K*

is called the idele class group of K .

The relation between the ideal class group C I K and the idele class group C K is as follows. There is a surjective homomorphism

( ): h -----+ JK, ex f-----+ (ex) = TI pVp(u p), pfoo

from the idele group h to the ideal group J K. Its kernel is

/100 = TI K; x TI Up. ploo pfoo


It induces a surjective homomorphism

CK ---+ CIK

with kernel 1ft) K* I K*. We may also consider the surjective homomorphism

h ---+ J (0), a 1-----+ n pvp(ap) , p

onto the replete ideal group J (0). Its kernel is

I~ = {(ap) E h I laplp = 1 for all p}

(see chap. ill, § 1). It takes principal ideles to replete principal ideals and induces a surjective homomorphism

CK ---+ Pic(o)

onto the replete ideal class group, with kernel I~K* I K*. We therefore have the

(1.3) Proposition. CIK ~ hllft)K*, andPic(o) ~ hll~K*.

In contrast to the ideal class group, the idele class group is not finite. But the finiteness of the former is reflected in terms of the latter as follows.

(1.4) Proposition. h = liK*, i.e., CK = liK* I K*, if S is a sufficiently big finite set of places of K .

Proof: Let a 1, ... , ah be ideals representing the h classes of J K I PK. They are composed of a finite number of prime ideals p 1, ... , Pn. Now if S is any finite set of places containing these primes and the places at infinity, then one has IK = liK*.

In order to see this, we use the isomorphism hi 1;00 ~ J K. If a E h, then the corresponding ideal (a) = npfoo pvp(ap) belongs to some class ai PK,

i.e., (a) = ai(a) for some principal ideal (a). The idele a' = aa-1 is mapped by h -+ J K to the ideal ai = npfoo p Vp (a~). Since the prime ideals occurring in ai lie in S, we have vp(a~) = 0, i.e., a~ E Up for all p ¢ S. Hence a' = aa-1 E Ii, and thus a E liK*. 0

§ 1. Idetes and Idele Classes 361

The ideIe group comes equipped with a canonical topology. A basic system of neighbourhoods of I ElK is given by the sets

TIWpxTIUpS;h, PES p\<,S

where S runs through the finite sets of places of K which contain all pi 00,

and Wp S; K; is a basic system of neighbourhoods of I E K;. The groups Up are compact for p f/. S. Therefore the same is true of the group TIp\<,s Up. If the Wp, for ploo, are bounded, then TIPES Wp x TIp\<,S Up is a neighbourhood of I in I K whose closure is compact. Therefore I K is a locally compact topological group.

(1.5) Proposition. K* is a discrete, and therefore closed, subgroup of h.

Proof: It is enough to show that I E h has a neighbourhood which contains no other principal idele besides 1.

II = { a E h I lap I p = I for p t 00, lap - 11 p < I for pi oo} ,

is such a neighbourhood. For if we had a principal idele x E II different from 1, then we get the contradiction

1 = TI Ix -lip = TI Ix -lip· TI Ix - lip p pfoo pioo

< TI Ix - lip::: TI maxI Ixlp, 1} = 1. pfoo pfoo

That the subgroup is closed follows for a completely general reason: since (x, y) r-+ x y -I is continuous, there is a neighbourhood V of I such that V V -I S; ll. For every y ElK, the neighbourhood y V then contains at most one x E K*. Indeed, from XI = yVI, X2 = yV2 E K*, with XI =j:. X2, one deduces xlxi l = VI ViI Ell, a contradiction. 0

As K* is closed in I K, the fact that I K is a locally compact Hausdorff topological group carries over to the idele class group C K = I K / K*. For any idele a = (ap) E h, its class in C K will be denoted by [a]. We define the absolute norm of a to be the real number

l)1(a) = TII)1(p)vp (ap) = TI lapl;l. p p

If x E K* is a principal idele, then we find by chap. III, (1.3), that l)1(x) = TIp Ixl;1 = 1. We thus have a continuous homomorphism

1)1 : CK ----+ lR:.


It is related to the absolute norm on the replete Picard group PicCo) via the commutative diagram

1 'J1

---+) ~~

'J1 PicCo) ----+ ~~.

Here the arrow

CK -+ PicCo)

is induced by the continuous surjective homomorphism

h -+ J(o), (ap) 1---+ n pvp(ap) , p

with kernel

If = { (ap) E h I lap I I' = 1 for all p} .

As to the kernel C~ of SJ1 : CK -+ ~+, we obtain, in analogy with chap. III, (1.14), the following important theorem. It reflects the finiteness of the unit rank of K as well as the finiteness of the class number.

(1.6) Theorem. The group C~ = Ha] E CK I SJ1([a]) = I} is compact.

Proof: The claim concerning the commutative exact diagram

1 ---+ C~ ---+) ~~ -+ 1

1 1 1 -+ Pic(o)o ----+ PicCo) ----+ ~~ -+ 1

will be reduced to the compactness of the group Pic(o)o, which was proved in chap. III, (1.14). The kernel of the vertical arrow in the middle is the group IfK* f K* = Iff If n K*, where we have If = np Ig,

Ig = tap E Kp I lap II' = I}, and If n K* = J.L(K) by chap. III, (1.9). This kernel is clearly compact. We obtain an exact sequence

1 -+ IfK* f K* -+ C~ -+ Pic(o)o -+ 1

of continuous homomorphisms. Since Pic(o)o is compact, and the same is true for the fibres of the mapping C~ -+ Pic(o)o (they are cosets, all homeomorphic to IfK*fK*), hence so is C~. 0


The idele class group C K plays a similar role for the algebraic number field K as the multiplicative group K; does for a p-adic number field Kp. It comes equipped with a collection of canonical subgroups which are to be viewed as analogues of the higher unit groups UJn) = I + pn of a p-adic number field Kp. Instead of pn, we take any integral ideal m = TIptoo pnp. We may also write it as a replete ideal

m = TIpn p

p

with np = 0 for pi 00, and we treat it in what follows as a module of K. For

every place p of K we put UJO) = Up, and

/

1 + pnp , if P too, U (n p)._ ffi.* c K* if h is real, p .- + p' I"

C* = K;, if p is complex,

for np > O. Given lXp E K; we write

lXp == 1 mod pnp <==} lXp E U~np)

For a finite prime p and np > 0 this means the usual congruence; for a real place, it symbolizes positivity, and for a complex place it is the empty condition.

(1.7) Definition. The group

CK = IKK* IK*,

formed from the idele group

1m - TI U(n p) K - p'

p

is called the congruence subgroup mod m, and the quotient group CK IC/( is called the ray class group mod m.

Remark: This definition of the ray class group does correspond to the classical one, as given (in the ideal-theoretic version) for instance in Hasse's "Zahlbericht" [53]. It differs from those found in modern textbooks, and also from that given in [107] by the author: in the present book, the components lXp of ideles lX in If{ are always positive at all real places p, so we have here fewer congruence subgroups than in the other texts. This choice does not only simplify matters. Most of all, it was made substantially because of the choice


of the canonical metric ( , ) on the Minkowski space KJR (see chap. I, §5). In fact, we saw in chap. III, § 3, that this choice forces the extension C 1lR. to be unramified. We will explain in § 6 below how to interpret this situation, and how to reconcile it with the definition of ray classes in other texts.

The significance of the congruence subgroups lies in that they provide an overview over all closed subgroups of finite index in C K. More precisely, we have the

(1.8) Proposition. The closed subgroups of finite index of C K are precisely those subgroups that contain a congruence subgroup Ci(.

Proof: Ci( is open in C K because IF = np U~np) is open in h. IF is contained in the group I~oo = np100 K; x n p100 Up, and since

(CK : I~oo K* jK*) = #CIK = h < 00, the index

(CK : Ci() = h(I~oo K* : IFK*) ~ h(I~oo : IF)

= h n (Up : U~np) n (K; : U~np) pfoo ploo

is finite. Being the complement of the nontrivial open cosets, which are finite in number, Ci( is closed of finite index. Consequently, every group containing Ci( is also closed of finite index, for it is the union of finitely many cosets of Ci(.

Conversely, let N be an arbitrary closed subgroup of finite index. Then N is also open, being the complement of a finite number of closed cosets. Thus the preimage J of N in h is also open, and it thus contains a subset of the form

W = n Wp x n Up, peS p¢S

where S is a finite set of places of K containing the infinite ones, and Wp is an open neighbourhood of 1 E K;. If PES is finite, we are liable to choose Wp = U~np), because the groups U~np) ~ K; form a basic system of neighbourhoods of 1 E K;. If PES is real, we may choose Wp ~ lR.~. The open set Wp will then generate the group lR.~, resp. K; in the case of a complex place p. The subgroup of J generated by W is therefore of the form IF, so N contains the congruence subgroup Ci(. 0


The ray class groups can be given the following purely ideal-theoretic description. Let J'K be the group of all fractional ideals relatively prime to m, and let P'K be the group of all principal ideals (a) E PK such that

a == 1 mod m and a totally positive.

The latter condition means that, for every real embedding K ~ lR, a turns out to be positive. The congruence a == 1 mod m means that a is the quotient b I c of two integers relatively prime to m such that b == c mod m. This is tantamount to saying that a == 1 mod pnp in K p, i.e., a E U~np) for all p 1m = TIpfoo pnp. We put

We then have the

(1.9) Proposition. The homomorphism

( ): h ~ JK, a r-+ (a) = TI pvp(ap) , Pfoo

induces an isomorphism

Proof: Let m = TIp pnp , and let

It) = {a E h lap E U~np) for Plmoo} .

Then IK = lim) K*, because for every a E IK, by the approximation theorem, there exists an a E K* such that apa == 1 mod pnp for p 1m, and apa > 0 for p real. Thus f3 = (apa) E I~m), so that a = f3a- 1 E I~m) K*.

The elements a E It) n K* are precisely those generating principal ideals in P'K. Therefore the correspondence a 1-* (a) = TIpfoo pvp(ap) defines a surjective homomorphism

CK = I~m) K* IK* = It) lIt) n K* ~ J'KIP'K.

Since (a) = 1 for a E I'K, the group CK = I'KK* I K* is certainly contained in the kernel. Conversely, if the class [a] represented by a E I~m) belongs

to the kernel, then there is an (a) E P'K, with a E It) n K*, such that (a) = (a). The components of the idele f3 = aa-1 satisfy f3p E Up for

p t moo, and f3p E U~np) for plmoo, in other words, f3 E I'K, and hence [a] = [f3] E I'KK*IK* = CK. Therefore CK is the kernel of the above mapping, and the proposition is proved. 0


The ray class groups in the ideal-theoretic version C IY( = rp / rp were introduced by HEINRICH WEBER (1842-1913) as a common generalization of ideal class groups on the one hand, and the groups (ZlmZ)* on the other. These latter groups may be viewed as the ray class groups of the field Q:

(1.10) Proposition. For any module m = (m) of the field Q, one has

CQ/CQ ~ C/(; ~ (ZlmZ)*.

Proof: Every ideal (a) E if; has two generators, a and -a. Mapping the positive generator onto the residue class mod m, we get a surjective homomorphism if; ---+ (ZlmZ)* whose kernel consists of all ideals (a) which have a positive generator == 1 mod m. But these are precisely the ideals (a) such that a == 1 mod pnp for plmoo, i.e., the kernel of Pf;. 0

The group (ZlmZ)* is canonically isomorphic to the Galois group G(Q(J.Lm)IQ) of the m-th cyclotomic field Q(J.Lm). We therefore obtain a canonical isomorphism

G(Q(J.Lm)IQ) ~ CQ/CQ. It is class field theory, which provides a far-reaching generalization of this important fact. For all modules m of an arbitrary number field K, there will be Galois extensions K m I K generalizing the cyclotomic fields: the so-called ray class fields, which satisfy canonically

G(KmIK) ~ CK ICI(

(see § 6). The ray class group mod 1 is of particular interest here. It is related to the ideal class group C I K - which according to our definition here, is in general not a ray class group - as follows.

(1.11) Proposition. There is an exact sequence

1 ---+ 0* /o~ ---+ TI JR* /JR~ ---+ clk ---+ CIK ---+ 1, P real

where o~ is the group of totally positive units of K .

Proof: One has C/k ~ CKICi = hlIkK* and, by (1.3), CIK

hIIft"K*, where Ik = TIpUp and I~oo = TIpfooUp x TIplooK;. We therefore obtain an exact sequence

1 ---+ I~oo K* IIkK* ---+ CK Ici ---+ CIK ---+ 1.


For the group on the left we have the exact sequence

1 ----+ I Soo n K*jIl n K* ----+ ISOOjI 1 ----+ I Soc K*jIl K* ----+ 1 KKK KKK .

But /100 n K* = 0*, Ik n K* = o~, and I~OOjIk = flp!OOK;jUp = flpreal~*j~~. 0

Exercise 1. (i) AQ = (Z Q9z Q) x K

(ii) The quotient group AQ/7l., is compact and connected.

(iii) AQ/7l., is arbitrarily and uniquely divisible, i.e., the equation nx = Y has a unique solution, for every n EN and Y E AQ/7l.,.

Exercise 2. Let K be a number field, m = 2"m' (m' odd), and let S be a finite set of primes. Let a E K* and a E K;m, for all P ¢ S. Show:

(i) If K (~2V ) I K is cyclic, where ~2v is a primitive 2" -th root of unity, then a E K*m. (ii) Otherwise one has at least that a E K*m/2.

Hint: Use the following fact, proved in (3.8): if LIK is a finite extension in which almost all prime ideals split completely, then L = K.

Exercise 3. Write Ik = Ii x I~, with Ii = Opfoo Up, I~ = OPIOO Up. Show that taking integer powers of ideles a E I/ extends by continuity to exponentiation aX

with x E Z. Exercise 4. Let el, ... ,et E (')~ be independent units. The images S I, .. ·,st in I/ are then independent units with respect to the exponentiation with elements of Z, i.e., any relation

S~l ... s;' = 1, Xi E 7l."

implies Xi = 0, i = 1, ... ,to

Exercise 5. Let e E (')~ be totally positive, i.e., eEl k. Extend the exponentiation 7l., -+ I k, n ~ en, by continuity to an exponentiation Z x JR -+ I k = Ii x I ~, e ~ eA, in such a way that SJt(eA) = 1.

Exercise 6. Let PI, ... , Ps be the complex primes of K. For Y E JR, let cf>k(Y) be the idele having component e2rriy at Pb and components 1 at all other places. Let el, ... , et be a 7l., -basis of the group of totally positive units of K.

(i) The ideles of the form

a = e~l ... e;t cf>1 (Yl) ... cf>s(Ys) , Ai E Z x JR, Yi E JR,

form a group, and have absolute norm SJt(a) = 1.

(ii) a is a principal ideal if and only if A/ E 7l., S; Z x JR and Yi E Z S; JR.

Exercise 7. Sending

(AI, ... ,At,YI, ... ,Ys) ~ e~I ... e;tcf>I(YI)···cf>s(Ys)

defines a continuous homomorphism

f : (Z x JR)t X JRs ---+ C~ into the group C~ = ([a] E CK I SJt([a]) = I}, with kemel7l.,t x 7l.,s.


Exercise 8. (i) The image D~ of f is compact, connected and arbitrarily divisible. (ii) f yields a topological isomorphism

f: «Z x R)jll)t x (RjIlY ~ D~.

Exercise 9. The group D~ is the intersection of all closed subgroups of finite index in e~, and it is the connected component of 1 in e~.

Exercise 10. The connected component D K of 1 in the idele class group e K is the direct product of t copies of the "solenoid" (Z x R)jll, s circles Rjll, and a real line.

Exercise 11. Every ideal class of the ray class group e l~ can be represented by an integral ideal which is prime to an arbitrary fixed ideal.

Exercise 12. Let 0 = OK. Every class in (ojm)* can be represented by a totally positive number in 0 which is prime to an arbitrary fixed ideal.

Exercise 13. For every module m, one has an exact sequence

1 -+ 0* jom ---+ (ojm)* ---+ elm ---+ ell -+ 1 + + K K '

where o~, resp. o~, is the group of totally positive units of 0, resp. of totally positive units = 1 mod m.

Exercise 14. Compute the kernels of el~ -+ elK and el~ -+ Cit for m'lm.

§ 2. Ideles in Field Extensions

We shall now study the behaviour of ideles and idele classes when we pass from a field K to an extension L. So let L I K be a finite extension of algebraic number fields. We embed the idele group h of K into the idele group h of L by sending an idele cx = (cxp) E h to the idele cx' = (cx~) E h whose components cx~ are given by

cx~ = cxp E K; ~ L~ for qJlp.

In this way we obtain an injective homomorphism

h-+h,

which will always be tacitly used to consider h as a subgroup of h. An element cx = (cx'+!) E h therefore belongs to the group h if and only if its components cx'+! belong to Kp (qJlp), and if one has furthermore cx'+! = cx\j3' whenever qJ and qJ' lie above the same place p of K.

Every isomorphism a : L --* a L induces an isomorphism

a: h -+ I(JL

§ 2. Ideles in Field Extensions

like this. For each place SlJ of L, a induces an isomorphism

a : LrJ,J ----+ (aL)arJ,J.

369

For if we have a = SlJ -lim aj, for some sequence ai E L, then the sequence aai E aL converges with respect to I larJ,J in (aL)arJ,J, and the isomorphism is given by

a = SlJ-lim ai t-+ aa = aSlJ-lim aai.

For an idele a E h, we then define aa E h to be the idele with components

(aa)arJ,J = aarJ,J E (aL)arJ,J'

If L IK is a Galois extension with Galois group G = G(L IK), then every a E G yields an automorphism a : h --+ h, i.e., h is turned into an G-module. As to the fixed module If = {a E h I aa = a for all a E G}, we have the

(2.1) Proposition. If L I K is a Galois extension with Galois group G, then

If=h.

Proof: Let a E h ~ h. For a E G, the induced map a : LrJ,J --+ LarJ,J is a K,,-isomorphism, if SlJlp. Therefore

(aa)arJ,J = aarJ,J = arJ,J = aarJ,J,

so that aa = a, and therefore a E If. If conversely a = (as;p) E If, then

(aa)arJ,J = aarJ,J = aarJ,J

for all a E G. In particular, if a belongs to the decomposition group GrJ,J = G(LrJ,JIK,,), then aSlJ = SlJ and aarJ,J = arJ,J so that arJ,J E K;. If a E G is arbitrary, then a : LrJ,J --+ LarJ,J induces the identity on K", and we get arJ,J = aarJ,J = aarJ,J for any two places SlJ and aSlJ above p. This shows that aEh. 0

The idele group h is the unit group of the ring of adeles AL of L. It is convenient to write this ring as

where

L" = TI LrJ,Jo rJ,J1"


The restricted product IJpLp consists of all families (ap) of elements ap E Lp such that ap E Op = TI'.lJIP O'.lJ for almost all p. Via the diagonal embedding

Kp ---+ L p,

the factor Lp is a commutative Kp-algebra of degree L'.lJIP[L'.lJ : Kp) = [L : K). These embeddings yield the embedding

AK ---+ AL,

whose restriction

h = Ai< '---+ At = h turns out to be the inclusion considered above.

Every ap E L~ defines an automorphism

a p : Lp ---+ L p, x ~ apx,

of the Kp-vector space L p, and as in the case of a field extension, we define the norm of ap by

NLpIKp(ap) = det(ap).

In this way we obtain a homomorphism

NLplKp : L; ---+ K;. It induces a norm homomorphism

NLIK : h ---+ h

between the ideIe groups h = IJpL~ and h = IJpK;. Explicitly the norm of an idele is given by the following proposition.

(2.2) Proposition. If L I K is a finite extension and a = (a'.lJ) E h, the local components of the idele NLIK(a) are given by

NLIK(a)p = TI NL<.pIKp(a'.lJ). '.lJlp

Proof: Putting ap = (a'.lJ)'.lJIP E L p, the Kp-automorphism ap : Lp -+ Lp is the direct product of the Kp-automorphisms a'.lJ : L'.lJ -+ L'.lJ. Therefore

NLpIKp(ap) = det(ap) = TI det(a'.lJ) = TI NL<.pIKp(a'.lJ). 0 '.lJlp '.lJlp

The idele norm enjoys the following properties.

§ 2. Ideles in Field Extensions 371

(2.3) Proposition. (i) For a tower of tields K ~ L ~ M we have

NMIK = NLIK 0 NMIL.

(ii) If L IK is embedded into the Galois extension MIK and ifG = G(MIK) and H = G(MIL), then one has fora E h: NL\K(a) = OUEG/H ua.

(iii) NL\K(a) = a[L:K] fora E h.

(iv) The norm of the principal idele x E L * is the principal idele of K detined by the usual norm NL\K(X).

The proofs of (i), (ii), (iii) are literally the same as for the norm in a field extension (see chap. I, §2). (iv) follows from the fact that, once we identify Lp = L ®K Kp (see chap. II, (8.3», the Kp-automorphism Ix : Lp ---+ Lp, y 1-+ xy, arises from the K -automorphism x : L ---+ L by tensoring with Kp. Hence det(fx) = det(x).

Remark: For fundamental as well as practical reasons, it is convenient to adopt a formal point of view for the above considerations which allows us to avoid the constant back and forth between ideles and their components. This point of view is based on identifying the ring of adeles AL of L as

AL =AK ®KL,

which results from the canonical isomorphisms (see chap. II, (8.3»

Kp ®K L ~ Lp = n L;p, ap ®a ~ ap' (rs.pa). <:JJ\P

Here rs.p denotes the canonical embedding rs.p : L ---+ L<:JJ.

In this way the inclusion by components I K ~ I L is simply given by the embedding AK "-+ AL, a 1-+ a ® 1, induced by K ~ L. An isomorphism L ---+ u L then yields the isomorphism

u : AL = AK ®K L ----+ AK ®K u L = AuL

via u(a ® a) = a ® ua, and the norm of an L-idele a E Ai, is simply the determinant

NL\K(a) = detAK(a)

of the endomorphism a : AL ---+ AL which a induces on the finite AKalgebra AL = AK ®K L.

Here are consequences of the preceding investigations for the idele class groups.


(2.4) Proposition. If L I K is a finite extension, then the homomorphism I K ---+ h induces an injection of idele class groups

CK---+CL, aK*f-----+aL*.

Proof: The injection h ---+ h clearly maps K* into L *. For the injectivity, we have to show that I K n L * = K*. Let M I K be a finite Galois extension with Galois group G containing L. Then we have I K ~ h ~ 1M, and

h n L * ~ h n M* S; (h n M*)G = h n M*G = h n K* = K*. 0

Via the embedding C K ---+ C L, the idele class group C K becomes a subgroup of C L: an element aL * E C L (a E h) lies in C K if and only if the class aL * has a representative a' in h. It is important to know that we have Galois descent for the idele class group:

(2.5) Proposition. If L IK is a Galois extension and G = G(L IK), then CL is canonically a G -module and C2 = C K.

Proof: The G-module h contains L * as a G-submodule. Hence every a E G induces an automorphism

CL~CL' aL*f-----+(aa)L*.

This gives us an exact sequence of G-modules

1 ---+ L * ---+ h ---+ C L ---+ 1.

We claim that the sequence

1 ---+ L *G ---+ If ---+ c2 ---+ 1 deduced from the first is still exact. The injectivity of L *G ---+ If is trivial. The kernel of If ---+ C2 is If n L* = h n L* = K* = L*G. The surjectivity of If ---+ C2 is not altogether straightforward. To prove it, let aL * E Cf, For every a E G, one then has a (aL *) = aL *, i.e., aa = aXa for some Xa E L *. This Xa is a "crossed homomorphism", i.e., we have

Xal' = Xa . a Xl' .

(fUJI (fUJI (f(){ T(){ (f(){ .. Indeed, X(fl' = a- = (f(){ . a- = a( a)a- = aXl'X(f. By HIlbert 90 III

Noether's version (see chap. IV, (3.8» such a crossed homomorphism is of the form Xa = ay /y for some y E L *. Putting a' = ay-l yields a'L * = aL * and aa' = aaay-l = axaay-l = ay-l = a', hence a' E If. This proves surjectivity. 0

§ 3. The Herbrand Quotient of the Idele Class Group 373

The norm map NLIK : h ~ h sends principal ideles to principal ideles by (2.3). Hence we get a norm map also for the ideIe class group C L,

NLIK : CL ----+ CK.

It enjoys the same properties (2.3), (i), (ii), (iii), as the norm map on the idele group.

Exercise 1. Let WI, •.. , Wn be a basis of L I K . Then the isomorphism L 0K K p ~ n \j3lp L \j3 induces, for almost all prime ideals j:J of K, an isomorphism

WIOpEj1···Ej1wno p ~ n O \j3, \j3lp

where op, resp. 0\j3, is the valuation ring of K p, resp. L\j3.

Exercise 2. Let L I K be a finite extension. The absolute norm S)1 of ideles of K, resp. L, behaves as follows under the inclusion iLIK : h ---7 h, resp. under the norm NLIK : h ---7 h:

S)1(iLIK(a» = S)1(a)[L:Kl for a E h,

S)1(NLIK (a» = S)1(a) for a E h.

Exercise 3. The correspondence between ideles and ideals, a f-+ (a), satisfies the following rule, in the case of a Galois extension L I K ,

(NLIK(a)) = NLIK«a)).

(For the norm on ideals, see chap. III, § 1.)

Exercise 4. The ideal class group, unlike the idele class group, does not have Galois descent. More precisely, for a Galois extension L I K, the homomorphism CIK ---7 Clr(LIK) is in general neither injective nor surjective.

Exercise S. Define the trace TrLIK : AL ---7 AK by TrLlK(a) = trace of the endomorphism x f-+ ax of the AK -algebra A L , and show:

(i) TrLIK(a)p = L\j3IP TrL\llIK p (a\j3).

(ii) For a tower of fields K <; L <; M, one has TrMIK = TrLIK 0 TrMIL.

(iii) If L I K is embedded into the Galois extension M I K, and if G = G (M I K) and H = G(MIL), then one has for a E A L, TrLIK(a) = LaEG/H O"a.

(iv) TrLIK(a) = [L : K]a for a E A K.

(v) The trace of a principal adele x E L is the principal adele in AK defined by the usual trace TrLIKCx).

§ 3. The Herbrand Quotient of the Idele Class Group

Our goal now is to show that the idele class group satisfies the class field axiom of chap. IV, (6.1). To do this we will first compute its Herbrand


quotient. It is constituted on the one hand by the Herbrand quotient of the idele group, and by that of the unit group on the other. We study the idele group first.

Let LIK be a finite Galois extension with Galois group G. The G-module h may be described in the following simple manner, which immediately reduces us to local fields. For every place p of K we put

L; = TI L~ and UL,p = TI Ur;p. r;plp r;plp

Since the automorphisms a E G permute the places of Labove p, the groups L: and UL,p are G-modules, and we have for the G-module h the decomposition

h = TIL;, p

where the restricted product is taken with respect to the subgroups U L, p ~ L:. Choose a place ~ of Labove p, and let Gr;p = G(Lr;pIKp) ~ G be its decomposition group. As a varies over a system of representatives of G/Gr;p, a~ runs through the various places of Labove p, and we get

L; = TIL~r;p = TIa(L~), UL,p = TIU(1r;p = TIa(Ur;p). (1 (1 (1 (1

In terms of the notion of induced module introduced in chap. IV, § 7, we thus get the following

(3.1) Proposition. L: and UL,p are the induced G-modu1es

L; = Indgq:l(L~), UL,p = Indgq:l(Ur;p).

Now let S be a finite set of places of K containing the infinite places. We

then define Ii = II, where S denotes the set of all places of L which lie above the places of S. For Ii we have the G -module decomposition

Ii = TI L; x TI UL,p, peS p¢S

and (3.1) gives the

(3.2) Proposition. If L IK is a cyclic extension, and if S contains all primes ramified in L, then we have for i = 0, - 1 that

Hi(G,Ii) ~ E9Hi(Gr;p,L~) and Hi(G,I£) ~ E9Hi(Gr;p,L~), peS P

where for each p, ~ is a chosen prime of Labove p.


Proof: The decomposition Ii = (EBpES L~) EB V, V = TIplis U L, P' gives us an isomorphism

Hi(G,li) = EB Hi(G,L~) EB Hi(G, V), PES

and an injection Hi(G, V) ~ TIplisHi(G,UL,P)' By (3.1) and chap. IV,

(7.4), we have the isomorphisms Hi(G,L~) ~ Hi(G<:fj,L'i:r;) and

Hi(G,UL,p) ~ Hi(G<:fj,U<:fj)' For p fj. S, L<:fjIKp is unramified. Hence Hi(G<:fj,U<:fj) = 1, by chap.V, (1.2). This shows the first claim of the proposition. The second is an immediate consequence:

Hi (G,ld = ~ Hi(G,If) ~ EBHi(G<:fj,L'i:r;)=EBHi(G<:fj,L'i:r;). S SPES P

o

The proposition says that one has H-\G, h) = {I}, because H-1 (G<:fj, L'i:r;) = {I} by Hilbert 90. Further it says that

h/NLIKh = EBK;/NL'llIKpL'i:r;, P

where \fl is a chosen place above p. In other words:

An idele a E h is a norm of an idete of L if and only if it is a norm locally everywhere, i.e., if every component ap is the norm of an element of L'i:r;.

As for the Herbrand quotient h(G, If) we obtain the result:

(3.3) Proposition. If L I K is a cyclic extension and if S contains all ramified primes, then

h(G, Ii) = TI np , PES

where np = [L<:fj : Kp].

Proof: We have H-1(G,lf) = TIPES H-1(G<:fj, L'i:r;) = 1 and

HO(G, Ii) = TI HO(G<:fj,L'i:r;). PES

By local class field theory, we find #Ho(G<:fj, L'i:r;) = (K; : NL'llIKpL'i:r;) = np' Hence

S #Ho(G,If) h(G I ) - - TI n

, L - #H-l(G,li) - PES p. o

Next we determine the Herbrand quotient of the G-module L S = L n If For this we need the following general


(3.4) Lemma. Let V be an s-dimensionallR-vector space, and let G be a finite group of automorphisms of V which operates as a permutation group on the elements of a basis VI, ••. , Vs: aVi = vu(i).

If r is a G-invariant complete lattice in V, i.e., a r 5; r for all a, then there exists a complete sublattice in r,

r' = ZWI + ... + ZWs ,

such that aWi = Wu(i) for all a E G.

Proof: Let I I be the sup-norm with respect to the coordinates of the basis VI, ••• , Vs. Since r is a lattice, there exists a number b such that for every x E V, there is ayE r satisfying

Ix - yl < b.

Choose a large positive number t E lR, and ayE r such that

ItVI - yl < b,

and define Wi = L ay, i = 1, ... ,s,

u(1)=i

i.e., the summation is over all a E G such that a(l) = i. For every LEG we then have

LWi = L Lay = L PY = W,(i). u(I)=i p(I)=,(i)

It is therefore enough to check the linear independence of the Wi. To do this, let s

LCiWi =0, Ci E lR. i=I

If not all of the Ci = 0, then we may assume ICi I ::::: 1 and Cj = 1 for some j. Let

y = tVI - y,

for some vector y of absolute value I y I < b. Then

Wi = L ay = t L vu(1) - Yi = tnivi - Yi, u(1)=i u(l)=i

where Iyd < gb, for g = #G, and ni = #{a E G I a(1) therefore get

with Iz I ::::: sgb, i.e.,

s s ° = L CiWi = t L Cinivi - Z, i=I i=I

z = tnjvj + L tCinivi. if.j

= i}. We

If t was chosen sufficiently large, then z cannot be written in this way. This contradiction proves the lemma. 0


Now let L IK be a cyclic extension of degree n with Galois group G = G(L IK), let S be a finite set of places containing the infinite places, and let 5 be the set of places of L that lie above the places of S. We denote

the group L S of 5 -units simply by L S .

(3.5) Proposition. The Herbrand quotient of the G -module L S satisfies

h(G,Ls) = ! TI np, n PES

where np = [L'.]3 : Kp].

Proof: Let {e'.]3 I s,p E 5} be the standard basis of the vector space V = TI'.]3d~. By (1.1), the homomorphism

'A: L S ---+ V, 'A(a) = L log lal'.]3e'.]3, '.]3ES

has kernel f.L(L) and its image is an (5" - I)-dimensional lattice, 5" = #5. We make G operate on V via

(Te'.]3 = eu '.]3·

Then 'A is a G-homomorphism because we have, for (T E G,

'A«Ta) = Llog l(Tal'.]3e'.]3 = Llog lal u-l'.]3(Teu-l'.]3 '.]3 '.]3

= (T(Llog laI U -l'.]3eU -l'.]3) = (T'A(a). '.]3

Therefore eo = L'.]3ES e'.]3 and 'A(L s) generate a G-invariant complete lattice r in V. Since Zeo is G-isomorphic to Z, the exact sequence

° ---+ Zeo ---+ r ---+ r /Zeo ---+ 0,

together with the fact that r /Zeo = 'A(L s), yields the identities

h(G, L S ) = h(G, 'A(L s» = h(G, Z)-lh(G, r) = !h(G, r). n

We now choose in r a sublattice r', in accordance with lemma (3.4). Then we have

r' = EBZw'.]3 = EB EB Zw'.]3 = EB r; '.]3 PES '.]3lp PES

and (Tw'.]3 = wu'.]3. This identifies r; as the induced G-module

r; = EB Zw'.]3 = EB (T (Zw'.]3o) = Indg P (Zw'.]3o) ' '.]3lp UEGjG p


where '.Po is a chosen place above p, and G p is its decomposition group. The lattice r' has the same rank as r, so is therefore of finite index in r. From chap. IV, (7.4), we conclude that

S 1 ,10 ,10 h(G, L ) = -h(G, r) = - h(G, rp) = - h(Gp, ZW\j3o) n n peS n peS

1 = - 0 h(Gp,Z).

n peS

Thus we do find that h(G, LS) = ~ OpeS np, where np = #Gp = [L\j3 : Kp]. o

From the Herbrand quotient of Ii and L S we immediately get the Herbrand quotient of the idele class group C L. To do it choose a finite set of places S containing all infinite ones and all primes ramified in L, such that h = Ii L *. Such a set exists by (1.4). From the exact sequence

1 ----+ L S ----+ Ii ----+ IiL * / L * ----+ 1

arises the identity

h(G,Cd = h(G,Il,)h(G,LS)-I,

and from (3.3) and (3.5) we obtain the

(3.6) Theorem. If L I K is a cyclic extension of degree n with Galois group G = G(LIK), then

h #Ho(G,Cd (G,Cd = #H-l(G,Cd = n.

In particular (CK : NLJKCd :::: n.

From this result we deduce the following interesting consequence.

(3.7) Corollary. If L IK is cyclic of prime power degree n = pV (v > 0), then there are infinitely many places of K which do not split in L.

Proof: Assume that the set S of nonsplit primes were finite. Let M I K be the subextension of L I K of degree p. For every p (j S, the decomposition group G p of L IK is different from G(L IK). Hence Gp ~ G(L 1M). Therefore every p (j S splits completely in M. We deduce from this that N M JK C M = C K , thus contradicting (3.6).


Indeed, let a E h. By the approximation theorem of chap. II, (3.4), there exists an a E K* such that apa-1 is contained in the open subgroup NM'-llIKpM~, for all pES. If P ¢:. S, then apa- 1 is automatically contained in NM'-llIKpM~ because M'lJ = Kp. Since

h jNMIK 1M = EB K;jNM'rJIKpM~, p

the idele aa- 1 is a norm of some idele fJ of 1M, i.e., a = (N MIK fJ)a E

N M I KIM K *. This shows that the class of a belongs to N M I K C ~, so that CK = NMIKCM. 0

(3.8) Corollary. Let L IK be a finite extension of algebraic number fields. If almost all primes of K split completely in L, then L = K.

Proof: We may assume without loss of generality that L IK is Galois. In fact, let M I K be the normal closure of L I K, and write G = G (M I K) and H = G(MIL). Also let p be a place of K, ~ a place of M above p, and let G'lJ be its decomposition group. Then the number of places of Labove p equals the number#H\GjG'lJ of double cosets HaG'lJ in G (see chap. I, §9). Hence p splits completely in L if #H\GjG'lJ = [L : K] = #H\G. But this is tantamount to G'lJ = 1, and hence to the fact that p splits completely in M.

So assume LIK is Galois, L # K, and let a E G(LIK) be an element of prime order, with fixed field K'. If almost all primes p of K were completely split in L, then the same would hold for the primes p' of K'. This contradicts (3.7). 0

Exercise 1. If the Galois extension L I K is not cyclic, then there are at most finitely many primes of K which do not split in L.

Exercise 2. If L I K is a finite Galois extension, then the Galois group G (L I K) is generated by the Frobenius automorphisms <J!\l3 of all prime ideals s,p of L which are unramified over K.

Exercise 3. Let L I K be a finite abelian extension, and let D be a subgroup of h such that K*D is dense in hand D ~ NLIKL*. Then L = K.

Exercise 4. Let L 1, ... , L r I K be cyclic extensions of prime degree p such that Lin L J = K for i #- j. Then there are infinitely many primes p of K which split completely in L i, for i > 1, but which are nonsplit in L 1.


§ 4. The Class Field Axiom

Having detennined the Herbrand quotient h (G , C £) to be the degree n = [L : K] of the cyclic extension L IK, it will now be enough to show either H- 1(G,C£) = 1 or HO(G,C£) = (CK : NLIKC£) = n. The first identity is curiously inaccessible by way of direct attack. We are thus stuck with the second. We will reduce the problem to the case of a Kummer extension. For such an extension the nonn group NLIKCL can be written down explicitly, and this allows us to compute the index (CK : NLIKC£).

So let K be a number field that contains the n-th roots of unity, where n is a fixed prime power, and let L IK be a Galois extension with a Galois group of the fonn

G(LIK) ~ (ZjnZr.

We choose a finite set of places S containing the ramified places, those that divide n, and the infinite ones, and which is such that h = IiK*. We write again K S = Ii n K* for the group of S-units, and we put s = #S.

(4.1) Proposition. One has s ::: r, and there exists a set T of s - r primes of K that do not belong to S such that

L = K(ZiLl) ,

Proof: We show first that L = K ( n) if ,,1 = L *n n K s, and then that ,,1 is the said kernel. By chap. IV, (3.6), we certainly have that L = K (YD), with D = L *n n K . If xED, then K p ( :.:.rx ) I K p is unramified for all p fj. S because S contains the places ramified in L. By chap. V, (3.3), we may therefore write x = upy;, with up E Up, YP E K;. Putting YP = 1 for PES, we get an idele Y = (yp) which can be written as a product y = az with a Eli, z E K*. Then xz-n = upa; E Up for all P fj. S, i.e., xz-n Eli n K* = K S,

so that xz-n E ,,1. This shows that D = LlK M , and thus L = K (n). The field N = K ( W) contains the field L because ,,1 = L *n n K S ~

K S • By Kummer theory, chap. IV, (3.6), we have

G(NIK) ~ Hom(KS j(KS)n,ZjnZ).

By (1.1), K S is the product of a free group of rank s - 1 and of the cyclic group f.L(K) whose order is divisible by n. Therefore K S j(KS)n

§ 4. The Class Field Axiom 381

is a free (ZjnZ)-module of rank s, and so is G(NIK). Moreover, G(NIK)jG(NIL) ~ G(LIK) ~ (ZjnZY is a free (ZjnZ)-module of rank r so that r ::::: s, and G(NIL) is a free (ZjnZ)-module of rank s -r. Let 0"1, ... ,O"s-r be a Z j nZ -basis of G (N I L), and let Ni be the fixed field of O"i ,

i = 1, ... , s - r. Then L = nr:~ Ni. For every i = 1, ... , s - r we choose a prime ~i of Ni which is nonsplit in N such that the primes PI, ... , Ps-r of K lying below ~I' ... , ~s-r are all distinct, and do not belong to S. This is possible by (3.7). We now show that the set T = {PI, ... ,Ps-r} realizes the group L1 = L*n n K S as the kernel of KS --+ TIpET K;jK;n.

Ni is the decomposition field of N I K at the unique prime ~; above ~i, for i = 1, ... ,s - r. Indeed, this decomposition field Zi is contained in Ni because ~i is non split in N. On the other hand, the prime Pi

is unramified in N, because by chap. V, (3.3), it is unramified in every extension K(!!/U), U E KS. The decomposition group G(NIZi) :2 G(NINd is therefore cyclic, and necessarily of order n since each element of G(NIK) has order dividing n. This shows that Ni = Zi.

From L = nr:~ Ni it follows that L I K is the maximal subextension of N I K in which the primes PI, ... , P s -r split completely. For x E K S we therefore have

x E L1 {=::} K($) ~ L {=::} K pi ($) = Kpi' i = 1, ... ,s -r, K *n· 1 {=::} x E Pi' l = , ... , s - r .

This shows that L1 is the kernel of the map K S --+ TIf':~ K;J K;;. 0

(4.2) Theorem. Let T be a set of places as in (4.1), and let

CdS,T) = h(S,T)K*jK*,

where

h(S,T) = TI K;n x TI K; x TI Up. PES pET p\iSuT

Then one has

NLIKCL:2 CK(S,T) and (CK: CK(S,T») = [L: K].

In particular, if LIK is cyclic, then NLIKCL = CdS, T).

Remark: It will follow from (5.5) that NLIKCL = CK(S, T) also holds in general.

For the proof of the theorem we need the following


(4.3) Lemma. h(S,T)nK* = (KSUT)n.

Proof: The inclusion (KSuT)n ~ h(S, T) n K* is trivial. Let y E

h(S, T) n K*, and M = K(~). It suffices to show that NM[KCM = CK, for then (3.6) implies M = K, hence y E K*n n h(S, T) ~ (KSuT)n. Let [a] E CK = I;K*/K*, and let a E I; be a representative of the class [a]. The map

is surjective. For if ..:1 denotes its kernel, then obviously K*n n ..:1 = (KS)n, and ..:1K*n/K*n = ..:1/(KS)n. From (1.1) and Kummer theory, we therefore get

nS ___ =ns- r .

#G(LIK)

This is also the order of the product because by chap. II, (5.8), we have #UP/U; = n since P f n. We thus find an element x E K S such that ap = xu~, up E Up, for pET. The idele a' = ax- l belongs to the same class as a, and we show that a' E NM[K 1M. By (3.2), this amounts to checking that every component a~ is a norm from Mq:tlKp. For PES this holds because y E K;n. Hence we have MqJ = Kp for pET since a~ = u~ is a n-th power. For P ¢ S U T it holds because a~ is a unit and MqJlKp is unramified (see chap. V, (3.3». This is why [a] E NMIKCM, q.e.d. 0

Proof of theorem (4.2): The identity (CK : CK(S, T» = [L : K] follows from the exact sequence

1 ~ I;UT n K* /h(S, T) n K* ~ I;UT /h(S, T)

~ I;UTK*/h(S,T)K* ~ 1.

Since I;UT K* = h, the order of the group on the right is

(I;uTK*: h(S,T)K*) = (hK*/K*: h(S,T)K*/K*)

= (CK : CK(S,T»).

The order of the group on the left is

(/;UT n K* : h(S, T) n K*) = (K SUT : (KSuT)n) = n2s - r

because #(S U T) = 2s - r, and ILn ~ KSUT . In view of chap. II, (5.8), the order of the group in the middle is

n2 (I;UT : h(S, T») = n (K; : K;n) = n - = n 2s n Inl;l = n2s.

PES PES In Ip p

§ 4. The Class Field Axiom 383

Altogether this gives

n2s

( CK : CK(S, T)) = -2- = nr = [L : K]. n s-r

We now show the inclusion CK(S, T) S; NLIKCL. Let Cl E h(S, T). In order to show that Cl E NLIK h all we have to check, by (3.2), is again that every component Clp is a norm from L\jJIKp. For PES this is true because Clp E K;n is an n-th power, hence a norm from Kp(~) (see chap. Y, (1.5», so in particular also from L\jJIKp. For pET it holds because (4.1) gives L1 S; K;n, and thus L\jJ = Kp. Finally, it holds for P ¢ S U T since Clp is a unit and L\jJIKp is unramified (see chap. Y, (3.3». We therefore have h(S, T) S; NLIK h, i.e., CK(S, T) S; NLIKCL.

Now if LIK is cyclic, i.e., ifr = 1, then from (3.6),

[L : K] :s (CK : NLIKCd :s (CK : CK(S, T») = [L : K],

hence NLIKCL = CK(S, T). D

Now that we have an explicit picture in the case of a Kummer field, the result we want follows also in complete generality:

(4.4) Theorem (Global Class Field Axiom). If L IK is a cyclic extension of algebraic number fields, then

#Hi (G(LIK), Cd = { [L : K] for ~ = 0, 1 fon = -1.

Proof: Since h(G(L IK), Cd = [L : K], it is clearly enough to show that #Ho(G(LIK),Cd I [L : K]. We will prove this by induction on the degree n = [L : K]. We write for short HO(LIK) instead of HO(G(LIK),Cd. Let M I K be a subextension of prime degree p. We consider the exact sequence

i.e., the exact sequence

HO(LIM) ~ HO(LIK) ~ HO(MIK) ~ 1.

If p < n, then #Ho(LIM) I [L : M], #Ho(MIK) I [M : K] by the induction hypothesis, hence #Ho(LIK) I [L : MUM: K] = [L : K].

Now let P = n. We put K' = K(JLp) and L' = L(JLp). Since d = [K' : K] I p - 1, we have G(LIK) ~ G(L'IK'). L'IK' is a cyclic


Kummer extension, so by (4.2), #Ho(L'IK') = [L' : K'J = p. It therefore suffices to show that the homomorphism

induced by the inclusion CL -+ Cu is injective. HO(LIK) has exponent p, because for x E CK we always have x P = NLIK(X). Taking d = [K' : KJ-th powers on HO(L IK) is therefore an isomorphism. Now let x = x mod NLIKCL belong to the kernel of (*). We write x = yd, for some y = y mod NLIKCL. Then y also is in the kernel of (*), i.e., y = NUIK'(Z'), z' E Cu, and we find:

i = NK'IK(Y) = NUIK(Z') = NLIK(NuILCz'» E NLIKCL.

Hence x = yd = 1. D

An immediate consequence of the theorem we have just proved is the famous Hasse Norm Theorem:

(4.5) Corollary. Let L IK be a cyclic extension. An element x E K* is a nonn if and only if it is a nonn locally everywhere, i.e., a nann in every completion LlJlKp (1.lJIp).

Proof: Let G = G(LIK) and G':jJ = G(L':jJIKp). The exact sequence

1 ---+ L * ---+ h ---+ C L ---+ 1

of G-modules gives, by chap. IV, (7.1), an exact sequence

H-1(G,Cd ---+ HO(G,L*) ---+ HO(G,h).

By (4.4), we have H-1(G, Cd = 1, and from (3.2) it follows that HO(G, h) = EBp HO(G':jJ, L~). Therefore the homomorphism

K* /NLIKL * ---+ EB K;/NLq:.!lKpL~ p

is injective. But this is the claim of the corollary. D

It should be noted that cyclicity is crucial for Hasse's norm theorem. In fact, whereas it is true by (3.2) that an element x E K* which is everywhere locally a norm, is always the norm of some ideIe ex of L, this need not be by any means a principal ideIe, not even in the case of arbitrary abelian extensions.

§ 5. The Global Reciprocity Law 385

Exercise 1. Detennine the nonn group NL1KCL for an arbitrary Kummer extension in a way analogous to the case treated in (4.2) where G(LIK) ~ (7!.,fpa71Y.

Exercise 2. Let ~ be a primitive m -th root of unity. Show that the nonn group NQ(OIQCQ equals the ray class group mod m = (m) in CQ.

Exercise 3. An equation X2 - ai = b, a, bE K*, has a solution in K if and only if it is solvable everywhere locally, i.e., in each completion K p.

Hint: X2 -ay2 = NK(JalIK(X - Jay) if art K*2.

Exercise 4. If a quadratic fonn a)x? + ... +anx; represents zero over a field K with more than five elements (i.e., a)x? + ... + anx; = 0 has a nontrivial solution in K), then there is a representation of zero in which all Xi =1= O.

Hint: If a~2 = A =1= 0, b =1= 0, then there are non-zero elements a and f3 such that aa2 + bf32 = A. To prove this, multiply the identity

(t - 1)2 4t ---+ =1 (t + 1)2 (t + 1)2

by a~2 = A and insert t = by2 la, for some element y =1= 0 such that t =1= ±l. Use this to prove the claim by induction.

Exercise 5. A quadratic fonn ax2 + bi + cz2, a, b, c E K*, represents zero if and only if it represents zero everywhere locally. Remark: In complete generality, one has the following "local-to-global principle":

Theorem of Minkowski-Hasse: A quadratic fonn over a number field K represents zero if and only if it represents zero over every completion Kp.

The proof follows from the result stated in exercise 5 by pure algebra (see [113]).

§ 5. The Global Reciprocity Law

Now that we know that the idele class group satisfies the class field axiom, we proceed to determine a pair of homomorphisms

obeying the rules of abstract class field theory as developed in chap. IV, § 4. For the i-extension of Q given by d, we have only one choice. It is described in the following

(5.1) Proposition. Let Q IQ be the field obtained by adjoining all roots of unity, and let T be the torsion subgroup of G(Q IK) (i.e., the group of all elements of finite order). Then the fixed field ij I Q of T is a i-extension.


Proof: Since Q = Un':':1 Q(J-tn), we find

G(QIQ) = U!!! G(Q(J-tn)IQ) = U!!! (Z/nZ)* = i*. n n

~

But Z = TIpZp, and Z; ~ Zp x Z/(p - I)Z for p #- 2 and Z~ ~ Z2 x Z/2Z. Consequently,

G(QIQ) ~ i* ~ i x f, where f = TI Z/(p -1)Z x Z/2Z. p#2

This shows that the torsion subgroup T of G (Q I Q) is isomorphic to the torsion subgroup of f. Since the latter contains the group EBp.t2 Z/(p - I)~ EB Z/2Z, we see that the closure f of T is iso~orphic to T. No~, if ~Q is the fixed field of T, this implies that G(Q IQ) G(QIQ)/T ~ Z. 0

Another description of the i-extension Q I Q is obtained in the following manner. For every prime number p, let QplQ be the field obtained by adjoining all roots of unity of p-power order. Then

G(QpIQ) = U!!! G(Q(J-tpv)IQ) = U!!! (Z/pVZ)* = Z;, v v

and Z; ~ Zp x Z/(p - I)Z for p #- 2 and Z~ ~ Z2 x Z/2Z. The torsion subgroup of Z; is isomorphic to Z/(p - I)Z, resp. Z/2Z, and taking its

fixed field gives an extension ijep) IQ with Galois group

G(ijep)IQ) ~ Zp,

~ ~ ~ ~ep)

The Z-extension QIQ is then the composite Q = TIp Q .

We fix an isomorphism G(ij IQ) ~ i. There is no canonical choice as in the case of local fields. However, the reciprocity law will not depend on the choice. Via G (ij I Q) ~ i, we obtain a continuous surjective homomorphism

d:GQ ~i

of the absolute Galois group GQ = G(QIQ). With this we continue as in chap. IV, §4, choosing k ~= Q as our base field. If K IQ is a finite extension, then we put fK = [K n Q : Q] and get a surjective homomorphism

I ~ dK = -d:GK ~Z,

fK

which defines the i-extension K = K ij of K. K I K is called the cyclotomic i-extension of K. We denote again by <[JK the element of G(K IK) which is


mapped to 1 by the isomorphism G(K IK) ~ Z, and by C{JLIK the restriction C{JK IL if L IK is a subextension of K I K. The automorphism C{JLIK must not be confused with the Frobenius automorphism corresponding to a prime ideal of L (see § 7).

For the GIQ-module A, we choose the union of the idele class groups C K of all finite extensions K I Q. Thus A K = C K. The henselian valuation v : C IQ ~ Z will be obtained as the composite

whe!,e the mapping [ ,Q I Q] will later turn out to be the norm residue symbol ( ,QIQ) of global class field theory (see (5.7)). For the moment we merely define it as follows.

For an arbitrary finite abelian extension L I K, we define the homomorphism

[ ,LIK]: h --+ G(LIK)

by

[a,LIK] = n(ap,LpIKp), p

where Lp denotes the completion of L with respect to a place 'PIp, and (ap, Lp IKp) is the norm residue symbol of local class field theory. Note that almost all factors in the product are 1 because almost all extensions Lp IKp are unramified and almost all ap are units.

(5.2) Proposition. If L I K and L' I K' are two abelian extensions of finite algebraic number fields such that K S; K' and L S; L', then we have the commutative diagram

IK' [ ,L'IK'J

) G(L'IK')

NK'IK 1 1 h

[ ,LIKJ ) G(LIK).

Proof: For an idele a = (a'.j3) E h' of K', we find by chap. IV, (6.4), that

(a'.j3,L~IK~)ILp = (NK~IKp(a'.j3),LpIKp), (~Ip),


and (2.2) implies

[NK'IK(a),LIK] = n(NK'IK(a)p, LplKp) = n n(NK, IK/a<:JJ),LpIKp) I' I' <:JJlp '.Jl

= n(a<:JJ,L~IK~)IL = [a,L'IK']IL· 0 <:JJ

If L I K is an abelian extension of infinite degree, then we define the homomorphism

[ ,LIK]: h ~ G(LIK)

by its restrictions [ ,LIK]lu:= [ ,L'IK] to the finite subextensions L' of L I K. In other words, if a E h, then the elements [a, L'I K] define, by (5.2), an element of the projective limit ~ G (L'I K), and [a, L I K] is

L' precisely this element, once we identify G(LIK) = ~ G(L'IK). Again one has the equation

[a,LIK] = n(ap,LpIKp), I'

where Lp does not denote the completion of L with respect to a place above p, but rather the localization, i.e., the union of the completions L~IKp of all finite subextensions (see chap. II, §8). Then LplKp is Galois, G(LpIKp) ~ G(LIK), and the product np(ap,LpIKp) converges in the profinite group to the element [a, L IK]. Indeed, if L'IK varies over the finite subextensions of LIK, then the sets Su = {p I (ap,L~IKp) =I- I} are all finite, so that we may write down the finite products

(Tu = n (aI" Lp IKp) E G(L IK). peSL'

They converge to [a, L IK], for if [a, L IK]G(L IN) is one of the fundamental neighbourhoods (i.e., N IK is one of the finite subextensions of L IK), then

(TL' E [a,LIK]G(LIN)

for all L' ;2 N because

(Tu IN = n(ap, NplKp) = [a,NIK] = [a,LIK] IN. I'

This shows that [a, L IK] is the only accumulation point of the family {(Tu}.

It is clear that proposition (5.2) remains true for infinite extensions L and L' of finite algebraic number fields K and K '.

§5. The Global Reciprocity Law 389

(5.3) Proposition. For every root of unity ~ and every principal idele a E K* one has

[a, K(~)IK] = 1.

Proof: By (5.2), we have [NKIQ(a), Q(~)IQ] = [a, K({)IK]IQ(~)' Hence we may assume that K = Q. Likewise we may assume that ~ has prime power order em =f. 2. Now let a E Q*, let vp be the normalized exponential valuation of Q for p =f. 00 and write a = uppvp(a). For p =f. e,oo, Qp({)IQp is unramified and (p, Qp({)IQp) is the Frobenius automorphism <{Jp : ~ -+ ~p. From chap. V, (2.4), we thus get

{ pvp(a) for p =f. e, 00,

(a,Qp(~)IQph = ~np with np = up! for p = e, sgn(a) for p = 00.

Hence

[a,Q({)IQR = IT(a,Qp({)IQp)~ = ~a p

where a = IT np = sgn(a) IT pvp(a)u"i! = sgn(a) IT pvp(a)a-! = 1. p P#,oo P#OO

o

Since the extension K I K is contained in the field of all roots of unity over K, the proposition implies

-[a,KIK] = 1

- ~ for all a E K*. The homomorphism [ ,KIK]: IK -+ G(KIK) therefore induces a homomorphism

and we consider its composite

VK : CK ----+ i

with dK : G(KIK) -+ i. The pair (dK, VK) is then a class field theory, for we have the

(5.4) Proposition. The map VK : C K -+ i is surjective and is a henselian valuation with respect to dK.


Proof: We first show surjectivity. If L I K is a finite subextension of K I K , then the map

[ , L IK) = TI( , LplKp) : h ---+ G(L IK) p

is surjective. Indeed, since ( ,LpIKp) : K; ~ G(LpIKp) is surjective, [h,LIK) contains all decomposition groups G(LpIKp). Thus all p split completely in the fixed field M of [h, L IK). By (3.8), this implies that M = K, and ~o [h, L IK) = G(-';JK). This yields furthermore that [h , K I K) = [C K , K I K) is dense in G (K I K). In the exact sequence

1 ---+ C~ ---+ C K ~ lR~ ---+ 1

(see § 1) the group C~ is compact by (1.6), and we obtain a splitting, if we identify lR~ with the group of positive real numbers in any infinite

completion Kp. ~us CK = C~ x lR~. Now, [lR~,KIK) = 1, for if x E lR~, then [x, K I K) I L = [x, L I K) = 1 for every finite subextension L I K of K I K, because we may always write x = yn with y E lR~ and n = [L : KL Therefore [CK,KIK) = [C~,KJK) is a closed, dense subgroup of G(K IK) and therefore equal to G(K IK). This proves the surjectivity of VK = dK 0 [ , K IK).

In the definition of a henselian valuation given in chap. IV, (4.6), condition (i) is satisfied because VK (C K) = Z, and condition (ii) follows from (5.2) because for every finite extension L I K we have the identity

vK(NLIKCd = vKCNLIK h) = ddNLIK h, K IK)

= iLIKddh,LIL) = iLlKVLCCd = iLIKZ. D

In view of the fact that the idele class group C K satisfies the class field axiom, the pair

constitutes a class field theory, ~the "global cl~s field theory". The above homomorphism v K = d K 0 [ , K I K) : C K ~ Z, for finite extensions K I Q, satisfies the formula

1 ~ 1 VK = -dQ 0 [ ,QIQ) ONKIQ = -vQ ONKIQ

fK fK

and is therefore precisely the induced homomorphism in the sense of the abstract theory in chap. IV, (4.7).

As the main result of global class field theory we now obtain the Artin reciprocity law:


(5.5) Theorem. For every Galois extension L IK of finite algebraic number fields we have a canonical isomorphism

ab ~ rLIK : G(LIK) ---+ CK/NLIKCL.

The inverse map of rLIK yields a surjective homomorphism

( ,LIK): CK ~ G(LIK)ab

with kernel N L IK C L. The map ( ,L I K) is called the global norm residue symbol. We view it also as a homomorphism h ~ G(LIK)ab.

For every place p of K, we have on the one hand the embedding G(LpIKp) "-+ G(L IK), and on the other the canonical injection

( ): K; ~ CK,

which sends ap E K; to the class of the idele

(ap) = ( ... ,1,1, l,ap, 1, 1, 1, ... ).

These homomorphisms express the compatibility of local and global class field theory, as follows.

(5.6) Proposition. If L I K is an abelian extension and p is a place of K, then the diagram

CK (,LIK)) G(LIK)

is commutative.

Proof: We first show that the proposition holds if L IK is a subextension of KII£, or if L = K(i);.J = p, and ploo. Indeed, the two maps [ ,KIK], ( ,KIK): h ~ G(KIK) agree because from chap. IV, (6.5), we have

- -dKO( ,KIK)=VK =dKO[ ,KIK]. -Thus, if L IK is a subextension of K IK and a = (ap) E h, then

(a,LIK) = [a,LIK] = TI(ap,LpIKp). p

In particular, for ap E K; we have the identity

«(ap),LIK) = (ap,LpIKp),

which shows that _the diagram is commutative when restricted to the finite subextensions of K I K .


On the other hand, let L = K(i), 1'100, and Lv =I- Kv. Then K; = JR*, JR~ is the kernel of ( ,LvIKv), and (-l,LvIKv) is complex conjugation in G(LvIKv) = G(CjJR). Thus, all we have to show is that «(-l),LIK) =I- 1. If we had «(-1), LIK) = 1, then the class of (-1) would be the nonn of a class of CL, i.e., (-l)a = NLIK(a) for some a E K* and an idele a E h. This would mean that a = NLqIK/aq) for q =I- I' and -a = NLpIKp(av), i.e., (a,LqIKq) = 1 for q =I- I' and (-a,LvIKv) = 1. By (5.3), we would have 1 = [a,LIK] = TIq(a,LqIKq) = (a,LvIKv), so that (-l,LvIKv) = 1, and therefore -1 E NLpIKp(L;) = NICiIRC* = lR~, a contradiction.

We now reduce the general case to these special cases as follows. Let L'IK' be an abelian extension, so that K ~ K', L ~ £I. We then consider the diagram

where Lv = KvL, K~ = KvK', L~ = KvL'. In this diagram, the top and bottom are commutative by chap. IV, (6.4), and the sides are commutative for trivial reasons. If now L'IK' is one of the special extensions for which the proposition is already established, then the back diagram is commutative, and hence also the front one, for all elements of G(LvIKv) in the image of G(L~IK~) -+ G(LvIKv). This makes it clear that it is enough to find, for every (j E G(LvIKv), some special extension £11K' such that (j lies in the image of G(L~ IK~). It is even sufficient to do this only for all (j of prime power order, because they generate the group. Passing to the fixed field of (j we may assume moreover that G(LIK) is generated by (j.

When 1'100 and Lv =I- K v, i.e., Kv = JR, Lv = C, we put £I = L(i) ~ C, and choose for K' the fixed field of the restriction of complex conjugation to £I. Then L' = K'(i) and K~ = JR, L~ = C, so the mapping G(L~IK~) -+ G(LvIKv) is surjective.

When I' f 00, we find the extension £11K' as follows. Let (j be of p-power order. ~We denote by K I K, resp. L I L, the Z p-extension contained in K I K , resp. L I L, and consider the field diagram


Lp-----Lp

L/f £/1 Kp -Rp

/ ~/ K K

with localizations Kp = KpK, Lp = LpL (all fields are considered to lie in a common bigger field). We may now lift (j E G(LpIKp) = G(LIK) to an automorphism fj of L p such that

(1) fj E G(LpIKp),

(2) fj I KA = ({JnA for some n EN. KIK

Indeed, since Kp = KpK # K p, the group 2(KpIKp) # 1, and thus is of finite index if viewed as subgroup of G(K IK) ~ 7l,p. It is jherefore generated by a natural power 1/1 = ({J~ K of Frobenius ({J ilK E G (K I K). As I __

in the proof of chap. IV, (4.4), we may then lift (j to a fj E G(LpIKp) such that fj I KA = 1/Im, mEN, so that fj I KA = ({JkAm •

p KIK

We now take the fixed field K' of fjli, and the extension L' = K'L. As in chap. IV, (4.5), conditions (ii) and (iii), it then follows that [K' : K] < 00

and K' = L. L'IK' is therefore a subextension of K'IK', and (j is the image of fj IL~ under G(L~ IK~) ~ G(Lp IKp). This finishes the proof. 0

(5.7) Corollary. If LIK is an abelian extension and a = (ap) E h, then

(a,LIK) = D(ap,LpIKp). p

In particular, for a principal idele a E K* we have the product formula

D(a, LplKp) = 1. p

Proof: Since h is topologically generated by the ideles of the form a = (ap), ap E K;, it is enough to prove the first formula for these ideles. But this is exactly the statement of (5.6):

(a,LIK) = «(ap),LIK) = (ap, LplKp) = D(aq,LqIKq). q

The product formula is a consequence of the fact that (a, L IK) depends only on the idele class a mod K * . 0


Identifying K; with its image in CK under the map ap 1-+ (ap), we obtain the following further corollary, where we use the abbreviations N = NLIK

and Np = NLpIKp.

(5.8) Corollary. For every finite abelian extension one has

NCL n K; = NpL;.

Proof: For xp E NpL; we see from (5.6) that «(xp),LIK) = (xp,LpIKp) = 1. Thus the class of (xp) is contained in NCL. Therefore NpL; ~ NCL. Conversely, let a E NCL n K;. Then a is represented on the one hand by a norm idele ex = N f3, f3 E h, and on the other hand by an idele (xp), xp E K;. This gives (xp)a = Nf3 with a E K*. Passing to components shows that a is a norm from L q I K q for every q =I p, and the product formula (5.7) shows that a is also a norm from LplKp. Therefore xp E NpL;, and this proves the inclusion NCL n K* ~ NpL;. 0

Exercise 1. If D K is the connected component of the unit element of C K, and if KablK is the maximal abelian extension of K, then CK/DK ~ G(KabIK).

Exercise 2. For every place p of K one has K~b = K ab K p.

Hint: Use (5.6) and (5.8).

Exercise 3. Let p be a prime number, and let M pi K be the maximal abelian pextension unramified outside of {p I p}. Further, let H I K be the maximal unramified subextension of M pi K in which the infinite places split completely. Then there is an exact sequence

1 -+ G(Mp IH) -+ G(Mp IK) -+ CIKCp) -+ 1, where CIKCp) is the p-Sylow subgroup of the ideal class group CIK, and there is a canonical isomorphism

G(MpIH) ~ [lU~l)/([lU~l)nE), pip pip

where E is the closure of the (diagonally embedded) unit group E = o~ in [lpiP Up.

Exercise 4. The group E(p) := E n [lpiP U~l) is a Zp-module of rank

rp(E) := rankzp(E(p» = [K : Ql] - rankzpG(MpIK). rp(E) is called the p-adic unit rank.

Problem: For the p-adic unit rank, one has the famous Leopoldt conjecture:

rp(E) = r + s - 1, where r, resp. s, is the number of all real, resp. complex, places; in other words,

rankzpG(MpIK) = s + 1. The Leopoldt conjecture was proved for abelian number fields K IQl by the American mathematician ARMAND BRUMER [22]. The general case is still open to date.

§ 6. Global Class Fields 395

§ 6. Global Class Fields

As in local class field theory, the reciprocity law provides also in global class field theory a complete classification of all abelian extensions of a finite algebraic number field K. For this it is necessary to view the idele class group C K as a topological group, equipped with its natural topology which the valuations of the various completions K'p impress upon it (see § 1).

(6.1) Theorem. The map

L 1---+ NL = NLIKCL

is a I-I-correspondence between the finite abelian extensions L I K and the closed subgroups of finite index in C K. Moreover one has:

L\ ~ L2 {=}NL1 2NL2' NL1L2 =NLI nNL2 , NL1nL2 =NL1N L2 .

The field L I K corresponding to the subgroup N of C K is called the class field of N. It satisfies

Proof: By chap. IV, (6.7), all we have to show is that the subgroups N of C K

which are open in the norm topology are precisely the closed subgroups of finite index for the natural topology.

If the subgroup N is open in the norm topology, then it contains a norm group NLIKCL and is therefore of finite index, because from (5.5), (CK : NLIKCd = #G(LIK)ab. To show that N is closed it is enough to show that NLIKCL is. For this, we choose an infinite place p of K and denote by rK the image of the subgroup of positive real numbers in K'p under the mapping ( ): K; --+ C K. Then rK is a group of representatives for the homomorphism sn : C K --+ 1R~ with kernel C~ (see § 1), i.e., C K = C~ x rK. By the same token, rK is a group of representatives for the homomorphism sn : C L --+ 1R~. We therefore get

NLIKCL = NLIKC2 x NLIKrK = NLIKC2 x r K = NLIKC2 x rK.

The norm map is continuous, and C2 is compact by (1.6). Hence N LIKC2 is closed. Since rK is clearly also closed in CK, we get that NLIKCL is closed.

Conversely let N be a closed subgroup of C K of finite index. We have to show that N is open in the norm topology, i.e., contains a norm group NLIKCL. For this we may assume that the index n is a prime power. For if n = p~l ... p:r , and Ni ~ C K is the group containing N of index p~i , then N = n~=\ Ni, and if the Ni are open in the norm topology, then so is N.


Now let J be the preimage of N with respect to the projection I K ---+ C K .

Then J is open in I K because N is open in C K (with respect to the natural topology). Therefore J contains a group

ul = n {l} x n Up, peS p¢S

where S is a sufficiently big finite set of places of K containing the infinite ones and those primes that divide n, such that h = IIK*. Since (h : J) = n, J also contains the group npES K;n x np¢s{l}, and hence the group

h(S) = n K;n x n Up. PES P¢S

Thus it is enough to show that C K (S) = h (S)K* / K * ~ N contains a norm group. If the n-th roots of unity belong to K, then CK(S) = NLIKCL with L = K ( ~ ), because of the remark following (4.2). If they do not belong to K, then we adjoin them and obtain an extension K'I K. Let S' be the set of primes of K' lying above primes in S. If S was chosen sufficiently large, then h, = Il:K'* and CK'(S') = NUIK'CU, with L' = K'( ':IK'S'), by the above argument. Using chap. V, (1.5), this gives on the other hand that NK'IK(lK'(S'» ~ h(S), so that

NUIKCU = NK'IK(NuIK'CU) = NK'IK(CK'(S'» ~ CK(S).

This finishes the proof. o

The above theorem is called the "existence theorem" of global class field theory because its main assertion is the existence, for any given closed subgroup N of finite index in C K, of an abelian extension L I K such that NLIKCL = N. This extension L is the class field for N. The existence theorem gives a clear overview of all the abelian extensions of K once we bring in the congruence subgroups CK of C K corresponding to the modules m = np100 pnp (see (1.7». They are closed of finite index by (1.8), and they prompt the following definition.

(6.2) Definition. The class field KmlK for the congruence subgroup CK is called the ray class field mod m.

The Galois group of the ray class field is canonically isomorphic to the ray class group mod m:


One has mim' ===> K m £ Knf ,

because clearly C}( ;:2 C1. Since the closed subgroups of finite index in C K

are by (1.8) precisely those subgroups containing a congruence subgroup C}(, we get from (6.1) the

(6.3) Corollary. Every finite abelian extension L IK is contained in a ray class field KmIK.

(6.4) Definition. Let L IK be a finite abelian extension, and let NL = NLIKCL. The conductor f of LIK (or of N£> is the gcd of all modules m such that L £ K m (i.e., C}( £ N£>.

K f I K is therefore the smallest ray class field containing L I K. But it is not true in general that m is the conductor of KmIK. In chap. V, (1.6), we defined the conductor f p of a p -adic extension L p I K p for a finite place p, to

be the smallest power fp = pn such that U~) £ NLpIKpL:. For an infinite place p we define fp = 1. Then we view f as the replete ideal f TIploo pO and obtain the

(6.5) Proposition. If f is the conductor of the abelian extension L I K , and fp is the conductor of the local extension L pi K p, then

f = TI fp· p

Proof: Let N = NLIKCL, and let m = TIppnp for pi 00). One then has

be a module (np = 0

C}( £ N {:=:} flm and TI fp 1m {:=:} fp Ipnp p

So to prove f = TIp f p, we have to show the equivalence

C}( £ N {:=:} fp Ipnp for all p.

It follows from the identity N n K; = NpL: (see (5.8)):

C}( £ N {:=:} (a E JK ::::} ii EN) for a ElK

for all p.

{:=:} (ap == 1 mod pnp ::::} (ap) EN n K; = NpL:) for all p

{:=:} (a E U(n p) ::::} a E N L *) {:=:} U(np) C N L * ..<----I.. f I"np p p p p p p - p p ~ Pt'·

o


By chap. V, (1.7), the local extension LplKp, for a finite prime p, is ramified if and only if its conductor f p is #- 1. This continues to hold also for an infinite place p, provided we call the extension L pi K p unramified in this case, as we did in chap. ID. Then (6.5) yields the

(6.6) Corollary. Let L IK be a finite abelian extension and f its conductor. Then:

P is ramified in L {=} pi f .

In the case of the base field Q, the ray class fields are nothing but the familiar cyclotomic fields:

(6.7) Proposition. Let m be a natural number and m = (m). Then the ray class field mod m of Q is the field

Qm = Q(JLm}

of m -th roots of unity.

Proof: Let m = TIp#oo pnp. Then Il; = TIp#oo U~np) x lR~. Let

m = m' pnp. Then U~np) is certainly contained in the norm group of the unramified extension Qp(JLml)IQp, but also in the norm group of Qp(JLpnp )IQp' according to chap. V, (1.8). This means, by §3, that every idele in Il; is a norm of some idele of Q(JLm). Thus CJ; ~ N CQ(JLm). On the other hand, CQ/CJ; ~ (Z/mZ)* by (1.10), and therefore

(CQ : CJ;) = [Q(JLm) : Q] = (CQ : NCQ(JLm») ,

so that CJ; = NCQ(JLm) , and this proves the claim. o

According to this proposition, one may view the general ray class fields KmlK as analogues of the cyclotomic fields Q(JLm) IQ. Nonetheless, they are not made to take over the important role of the latter because all we know about them is that they exist, but not how to generate them. In the case of local fields things were different. There the analogues of the ray class fields were the Lubin-Tate extensions which could be generated by the division points of formal groups - a fact that carries a long way (see chap. V, §5). This local discovery does, however, originate from the problem of generating global class fields, which will be discussed at the end of this section.

Note in passing that the above proposition gives another proof of the theorem of Kronecker and Weber (see chap. V, (loW)) to the effect that


every finite abelian extension L IQ is contained in a field Q(JLm) IQ, because by (1.8) the nonn group NLIQCL lies in some congruence subgroup C5, m = (m), so that L S; Q(JLm).

Among all abelian extensions of K, the ray class field mod 1 occupies a special place. It is called the big Hilbert class field and has Galois group

G(K1IK) ~ Clk .

By (1.11), the group Clk is linked to the ordinary ideal class group by the exact sequence

1 ---* 0* / o~ ---* TI ~ * /~~ ---* C I k ---* C I K ---* 1. p real

The big Hilbert class field has conductor f = 1 and may therefore be characterized by (6.6) in the following way.

(6.8) Proposition. The big Hilbert class field is the maximal unramified abelian extension of K .

Since the infinite places are always unramified, this means that all prime ideals are unramified. The Hilbert class field, or more precisely, the "small Hilbert class field", is defined to be the maximal unramified abelian extension HI K in which all infinite places split completely, i.e., the real places stay real. It satisfies the

(6.9) Proposition. The Galois group of the small Hilbert class field H I K is canonically isomorphic to the ideal class group:

G(HIK) ~ CIK .

In particular, the degree [H : K] is the class number hK of K.

Proof: We consider the big Hilbert class field KIIK and, for every infinite place p, the commutative diagram (see (5.6»

h/I}K*


The small Hilbert class field H 1 K is the fixed field of the subgroup Goo generated by all G(K~IKj:I)' 1'100. Under ( ,K1IK) this is the image of

( TI K;) IkK* /lkK* = 1;'00 K* /lkK*, j:lloo

where 1;'00 = TIj:lloo K; x TIJ:l'too Uj:I. Therefore by (1.3),

G(HIK) = G(K1IK)/Goo ~ h/l;,ooK* ~ CIK . 0

Remark: The small Hilbert class field is in general not a ray class field in terms of the theory developed here. But it is in many other textbooks where ray class groups and ray class fields are defined differently (see for instance [107]). This other theory is obtained by equipping all number fields with the Minkowski metric

(X,Y)K = LaTxTYT (T E Hom(K, CC)) , T

aT = 1 if T = f, aT = ! if T =I- f. A ray class group can then be attached to any replete module

m = npn p ,

j:I

where nj:l E Z, nj:l :::: 0, and nj:l = ° or = 1 if 1'100. The groups U~np) attached to the metrized number field (K, ( , ) K) are defined by

for nj:l > 0, and Uj:I for nj:l = 0, if l' f 00,

if l' is real and nj:l = 0,

if pis real and nj:l = I,

if l' is complex.

The congruence subgroup mod m of (K, ( , ) K) is then the subgroup CK = I/{K*/K* ofCK formed with the group

1 m - TI U(n p) K - j:I'

j:I

and the factor group C K / CK is the ray class group mod m. The ray class field mod m of (K, ( , ) K) is again the class field of K corresponding to the group CK ~ C K . As explained in chap. III, § 3, the infinite places l' have to be considered as ramified in an extension L IK if Lj:I =I- Kj:I. Likewise,


the conductor of an abelian extension L IK, i.e., the gcd of all modules m = ITp pnp such that CK ~ NLIKCL, is the replete ideal

f = IT fp, p

where now for an infinite place p, we have fp = pnp with np = 0 if Lp = K p,

and np = I if Lp #- Kp. Corollary (6.6) then continues to hold: a place p is ramified in L if and only if p occurs in the conductor f.

This entails the following modifications of the above theory, as far as ray class fields are concerned. The ray class field mod I is the small Hilbert class field. It is now the maximal abelian extension of K which is unramified at all places. The big Hilbert class field is the ray class field for the module m = ITploo p. In the case of the base field Q, the field Q (n of m -th roots of unity is the ray class field mod mpoo, where Poo is the infinite place. The ray class field for the module m becomes the maximal real subextension Q (~ + ~ -1 ), which was not a ray class field before. This is the theory one finds in the textbooks alluded to above. It corresponds to the number fields with the Minkowski metric. The theory of ray class fields according to the treatment of this book is forced upon us already by the choice of the standard metric (x, y) = Lr Xr Y r on KJR taken in chap. I, § 5. It is compatible with the Riemann-Roch theory of chap. III, and has the advantage of being simpler.

Over the field Q, the ray class field mod (m) can be generated, according to (6.7), by the m-th roots of unity, i.e., by special values of the exponential function e2rr: i z. The question suggested by this observation is whether one may construct the abelian extensions of an arbitrary number field in a similarly concrete way, via special values of analytic functions. This was the historic origin of the notion of class field. A completely satisfactory answer to this question has been given only in the case of an imaginary quadratic field K. The results for this case are subsumed under the name of Kronecker's Jugendtraum (Kronecker's dream of his youth). We will briefly describe them here. For the proofs, which presuppose an in-depth knowledge of the theory of elliptic curves, we have to refer to [96] and [28].

An elliptic curve is given as the quotient E = C / r of C by a complete lattice r = ZWI + Z~ in C. This is a torus which receives the structure of an algebraic curve via the Weierstrass p-function

1 [1 1 ] fJ (z) = fJr (z) = 2 + L ( )2 - 2 ' Z (]JEri Z -w w

where r ' = r " {OJ. fJ(z) is a meromorphic doubly periodic function, i.e.,

fJ (z + w) = fJ (z) for all w E r,


and it satisfies, along with its derivative ,9'(z), an identity

The constants g2, g3 only depend on the lattice r, and are given by g2 = g2(r) = 60 Lw#o ~4' g3 = g3(r) = 140 Lw#o ~6. ,9 and ,9' may thus be interpreted as functions on <C I r. If one takes away the finite set S ~ <CI r of poles, one gets a bijection

onto the affine algebraic curve in <c2 given by the equation y2 = 4x3 - g2X - g3. This gives the torus <CI r the structure of an algebraic curve E over <C of genus 1. An important-role is played by the j-invariant

2633g3 j(E) = j(r) = __ 2

L1 with L1 = g~ - 27g~.

It determines the elliptic curve E up to isomorphism. Writing generators WI, W2 of r in such an order that T = WI I W2 lies in the upper halfplane lHI, then j(E) becomes the value jeT) of a modular function, i.e., of a holomorphic function j on lHI which is invariant under the substitution

aT + b . (a b) T ~ CT + d for every matrix c d E SL2(71..).

Now let K ~ <C be an imaginary quadratic number field. Then the ring OK of integers forms a lattice in <C, and more generally, any ideal a of OK

does as well. The tori <C I a constructed in this way are elliptic curves with complex multiplication. This means the following. An endomorphism of an elliptic curve E = <C I r is given as multiplication by a complex number z such that zr ~ r. Generically, one has End(E) = 71... If this is not the case, then End(E) ® Ql is necessarily an imaginary quadratic number field K, and one says that this is an elliptic curve with complex multiplication. The curves <C I a are obviously of this kind.

The consequences of these analytic investigations for class field theory are the following.

(6.10) Theorem. Let K be an imaginary quadratic number field and a an ideal of OK. Then one has:

(i) The j-invariant j(a) of <CIa is an algebraic integer which depends only on the ideal class jt of a, and will therefore be denoted by j (jt).

(ii) Every j (a) generates the Hilbert class field over K.


(iii) If aI, ... , ah are representatives of the ideal class group elK, then the numbers j (ai) are conjugate to one another over K.

(iv) For almost all prime ideals p of K one has

wheref{Jp E G(K (j(a»!K) is the Frobenius automorphism ofaprime ideal s,p of K(j(a» above p.

Note that for a totally imaginary field K there is no difference between big and small Hilbert class field. In order to go beyond the Hilbert class field, i.e., the ray class field mod 1, to the ray class fields for arbitrary modules m =1= 1, we form, for any lattice r s; C, the Weber function

1_2735 g:f3 Pr(Z) , if g2g3 =1= 0,

rr(z)= -29362~P}(Z)' ifg2 =O,

2834 ~ p}(Z) , if g3 = O.

Let Ji Eel K be an ideal class chosen once and for all. We denote by Ji* the classes in the ray class group elK = [iF/Pi( which under the homomorphism

elK ~ elK

are sent to the ideal class (m)Ji- I . Let a be an ideal in Ji, and let b be an integral ideal in Ji*. Then abm- l = (a) is a principal ideal. The value ra(a) only depends on the class Ji*, not on the choice of a, b and a. It will be denoted by

With these conventions we then have the

(6.11) Theorem. (i) The invariants r(Jii), r(J't2), ... , for a fixed ideal class Ji, are distinct algebraic numbers which are conjugate over the Hilbert class field KI = K(j(Ji».

(ii) For an arbitrary Ji*, the field K (j (Ji), r (Ji*» is the ray class field mod m over K:


Exercise 1. Let K 'I K be the big, and HI K the small Hilbert class field. Then G(K'IH) ~ (7l./27l.y-t, where r is the number of real places, and 2t = (0* : o~).

Exercise 2. Let d > 0 be squarefree, and K = Q(.J(i). Let e be a totally positive fundamental unit of K. Then one has [K' : H] = 1 or = 2, according as NK1Q(e) = -lor = 1.

Exercise 3. The group (C K)n = (h)n K * / K * is the intersection of the nonn groups NL1KCL of all abelian extensions LIK of exponent n.

Exercise 4. (i) For a number field K, local Tate duality (see chap. V, § 1, exercise 2) yields a non-degenerate pairing

(*) IT H'(K p,7l./n7l.) x ITH'(Kp,tLn) -+ 7l./n7l. p p

of locally compact groups, where the restricted products are taken with respect to the subgroups H;'r(K p,7l./n7l.), resp. H;'r(Kp,tLn). For X = (Xp) in the first and a = (ap) in the second product, it is given by

(X,a) = LXp(ap,KpIKp). p

(ii) If L I K is a finite extension, then one has a commutative diagram

7l. / n7l.

IT H'(K p,7l./n7l.) x IT H'(Kp,tLn) ---+ 7l./n7l.. p p

(iii) The images of

and

H'(K,7l./n7l.) -+ ITH'(K p,7l./n7l.) p

H'(K,tLn) -+ IT H'(Kp,tLn) p

are mutual orthogonal complements with respect to the pairing (*).

Hint for (iii): The cokernel of the second map is C K / (C K)n, and one has H'(K,7l./n7l.) = Hom(G(LIK), 7l./n7l.) , where LIK is the maximal abelian extension of exponent n.

Exercise 5 (Global Tate Duality). Show that the statements of exercise 4 extend to an arbitrary finite GK-module A instead of 7l./n7l., and A' = Hom(A,K*) instead of tLn.

Hint: Use exercises 4--8 of chap. IV, §3, and exercise 4 of chap. V, § 1.

Exercise 6. If S is a finite set of places of K , then the map

H'(K,7l./n7l.) -+ IT H'(K p,7l./n7l.) PES

is surjective if and only if the map

H'(K,tLn) -+ IT H'(Kp,tLn) p¢S

§ 7. The Ideal-Theoretic Version of Class Field Theory 405

is injective. This is the case in particular if either the extension K (JL2" ) 1 K is cyclic, n = 2" m, (m, 2) = 1, or if S does not contain all places p 12 which are nonsplit in K (JL2") (see § 1, exercise 2).

Exercise 7 (Theorem of GRUNWALD). If the last condition of exercise 6 is satisfied for the triple (K, n, S), then, given cyclic extensions LplKp for PES, there always exists a cyclic extension L 1 K which has L p 1 K p as a completion for PES, and which satisfies the identity of degrees

[L : K] = scm{[Lp : Kp]}

(see also [10], chap. X, § 2).

Note: Let G be a finite group of order prime to #JL(K), let S be a finite set of places, and let LplKp, PES, be given Galois extensions whose Galois groups Gp can be embedded into G. Then there exists a Galois extension L 1 K which on the one hand has Galois group isomorphic to G, and which on the other hand has the given extensions LplKp as completions (see [109]).

§ 7. The Ideal-Theoretic Version of Class Field Theory

Class field theory has found its idele-theoretic fonnulation only after it had been completed in the language of ideals. From the very start, it was guided by the desire to classify all abelian extensions of a number field K. But at first, instead of the idele class group e K, there was only the ideal class group elK at hand to do this, along with its subgroups. In tenns of the insights that we have gained in the preceding section, this means the restriction to the subfields of the Hilbert class field, i.e., to the unramified abelian extensions of K. If the base field is Q, this restriction is of course radical, for Q has no unramified extensions at all by Minkowski's theorem. But over Q, we naturally encounter the cyclotomic fields Q(/Lm)IQ with their familiar isomorphisms G(Q(/Lm)IQ) ~ (ZjmZ)*. HEINRICH WEBER

realized, as was already mentioned, that the groups elK and (ZjmZ)* are -with a grain of salt - only different instances of a common concept, that of a ray class group, which he defined in an ideal-theoretic way as the quotient group

ezt; = ';jP; of all ideals relatively prime to a given module m, by the principal ideals (a) with a == 1 mod m, and a totally positive. He conjectured that this group e zt;, along with its subgroups, would do the same for the subextensions of a "ray class field" KmlK (which at first was only postulated to exist) as the ideal class group elK and its subgroups did for the subfields of the Hilbert class field. Moreover, he stated the hypothesis that every abelian


extension ought to be captured by such a ray class field, as was suggested by the case where the base field is Q, whose abelian extensions are all contained in cyclotomic fields Q(Jlm) IQ by the Kronecker-Weber theorem-. After the seminal work of the Austrian mathematician PHILIPP FURTWANGLER

[44], these conjectures were confirmed by the Japanese arithmetician TEm TAKAGI (1875-1960), and cast by EMIL ARTIN (1898-1962) into a definite, canonical form.

The idele-theoretic language introduced by CHEVALLEY brought the simplification that the idele class group C K encapsulated all abelian extensions of L IK at once, avoiding choosing a module m every time such an extension was given, in order to accommodate it into the ray class field KmIK, and thereby make it amenable to class field theory. The classical point of view can be vindicated in terms of the idele-theoretic version by looking at congruence subgroups Cj( in C K, which define the ray class fields K m 1 K. Their subfields correspond, according to the new point of view, to the groups between Cj( and C K, and hence, in view of the isomorphism

CK/Cj( ~ Clf(,

to the subgroups of the ray class group C If(.

In what follows, we want to deduce the classical, ideal-theoretic version of global class field theory from the idele-theoretic one. This is not only an obligation towards history, but a factual necessity that is forced upon us by the numerous applications of the more elementary and more immediately accessible ideal groups.

Let L 1 K be an abelian extension, and let p be an unramified prime ideal of K and qJ a prime ideal of L lying above p. The decomposition group G(L<:JJIKp) S; G(LIK) is then generated by the classical Frobenius automorphism

where Jr p is a prime element of K p. As an automorphism of L, qJp is obviously characterized by the congruence

qJpa == a q mod qJ for all a E (') L

where q is the number of elements in the residue class field of p. We put


Now let m be a module of K such that L lies in the ray class field mod m. Such a module is called an module of definition for L. Since by (6.6) each prime ideal p f m is unramified in L, we get a canonical homomorphism

(LIK) : r; ----+ G(LIK)

from the group r; of all ideals of K which are relatively prime to m by putting, for any ideal a = TIp P Vp :

( L I K ) = n ( L I K ) Vp •

a p P

( L~K) is called the Artin symbol. If pEr; is a prime ideal and il'p a prime element of K p, then clearly

( LIK) -P- = ({il'p},LIK) ,

if {il'p} E CK denotes the class of the idele ( ... ,1,1, il'p, 1, 1, ... ).

The relation between the idele-theoretic and the ideal-theoretic formulation of the Artin reciprocity law is now provided by the following theorem.

(7.1) Theorem. Let L IK be an abelian extension, and let m be a module of definition for it. Then the Artin symbol induces a surjective homomorphism

(LIK) : C1K ----+ G(LIK)

with kernel H m / PK, where H m = (NLIKJ;:)PK, and we have an exact commutative diagram

1 ----+ NLIKCL CK ( ,LIK)

) G(LIK) ----+ 1 -----)-

1 1 ( LIK) }d

1 ----+ H m / PK -----)- C1K ) G(LIK) ----+ 1.

Proof: In § 1, we obtained the isomorphism ( ) : C K / CK ---+ C lK = JK / P K by sending an idele a = (ap) to the ideal (a) = TIpfoo pVp(Clp). This isomorphism yields a commutative diagram

( ,LIK) G(LIK)

}d f elK ) G(LIK),

and we show that f is given by the Artin symbol.


Let P be a prime ideal not dividing m, Jr p a prime element of K p,

and c E CK /CK the class of the idele (Jrp) = ( ... ,I, I, Jrp,l,l, ... ). Then (c) = p mod PJ( and

( LIK) f(c») = (c,LIK) = ((Jrp},LIK) = -p- .

This shows that f : JJ( / PJ( ~ G(L IK) is induced by the Artin symbol (LIK) : JJ( ~ G(LIK), and that it is surjective.

It remains to show that the image of NLIKCL under the map ( ) : CK ~ JJ(/PJ( is the group Hm/PJ(. We view the module m = npfoo pnp as a module of L by substituting for each prime ideal p of K the product p = n'13IP ~e\ll!p. As in the proof of (1.9), we then get

CL = Il,m)L*/L*, where It) = {a E h I a'13 E U~\ll!pnp) for ~Imoo}. The elements of

NLIKCL = NLIK(/l,m»K* / K*

are the classes of norm ideles N L I K (a), for a E It). As

NLlda)p = n NL\llIKp(a'13) '13lp

(see (2.2», and since vp (N L\llIKp (a'13» = f'13IP v'13(a'13) (see chap. III, (1.2», the idele NLIK(a) is mapped by ( ) to the ideal

(NLlda» = n n pf\ll!pv\ll(a\ll) = NLIK( n ~v\ll(a\ll»). pfoo '131 p '13foo

Therefore the image of NLIKCL under the homomorphism ( ) : CK ~ JJ(/PJ( is precisely the group (NLIKJ"L)PJ(/PJ(, q.e.d. D

(7.2) Corollary. The Artin symbol ( L~K), for a E JI<, only depends on the class a mod P J(. It defines an isomorphism

(LIK) : JI</Hm ~ G(LIK).

The group H m = (NLIK J"L)PI< is called the "ideal group defined mod m" belonging to the extension L I K. From the existence theorem (6.1), we see that the correspondence L f-+ H m is 1-1 between subextensions of the ray class field mod m and subgroups of JI< containing PI<.

The most important consequence of theorem (7.1) is a precise analysis of the kind of decomposition of any unramified prime ideal p in an abelian extension LIK. It can be immediately read off the ideal group H m ~ JI< which determines the field L as class field.


(7.3) Theorem (Decomposition Law). Let LIK be an abelian extension of degree n, and let I' be an unramified prime ideal. Let m be a module of definition for L I K that is not divisible by I' (for instance the conductor), and let H m be the corresponding ideal group.

If f is the order ofp mod H m in the class group F; / H m , i.e., the smallest positive integer such that

pi E H m ,

then I' decomposes in L into a product

I' = ~l" '~r

of r = n / f distinct prime ideals of degree f over p.

Proof: Let I' = ~1 .•. ~r be the prime decomposition of I' in L. Since I' is unramified, the ~i are all distinct and have the same degree f. This degree is the order of the decomposition group of ~i over K, i.e., the order of the Frobenius automorphism ({Jp = ( L~K). In view of the isomorphism p;/Hm ~ G(LIK), this is also the order of p mod Hm in F;/Hm. This finishes the proof. 0

The theorem shows in particular that the prime ideals which split completely are precisely those contained in the ideal group Hf, if f is the conductor of L I K .

Let us highlight two special cases. If the base field is K = Ql and we look at the cyclotomic field Ql(JLm)IQl, the conductor is the module m = (m), and the ideal group corresponding to Ql(JLm) in Jl; is the group Pl;. As Jl; / Pl; ~ (Z/mZ)* (see (1.10», we obtain for the decomposition of rational primes p f m, the law which we had already deduced in chap. I, (10.4), and in particular the fact that the prime numbers which split completely are characterized by

p == 1 modm.

In the case of the Hilbert class field L I K, i.e., of the field inside the ray class field mod 1 in which the infinite places split completely, the corresponding ideal group H S; J k = J K is the group PK of principal ideals (see (6.9». This gives us the strikingly simple

(7.4) Corollary. The prime ideals of K which split completely in the Hilbert class field are precisely the principal prime ideals.


Another highly remarkable property of the Hilbert class field is expressed by the following theorem, known as the principal ideal theorem.

(7.5) Theorem. In the Hilbert class field every ideal Cl of K becomes a principal ideal.

Proof: Let K 11 K be the Hilbert class field of K and let K 21 K 1 be the Hilbert class field of K 1. We have to show that the canonical homomorphism

JK/PK ---+ JKl/PKI

is trivial. By chap. IV, (5.9), we have a commutative diagram

JKl/PKI - CKl/NK2IKICK2 G(K2IK1)

T Ti Tver = G(KtlK),

where i is induced by the inclusion C K ~ C K I. It is therefore enough to show that the transfer

Ver: G(KIIK) ---+ G(K2IK1)

is the trivial homomorphism. Since K 1 I K is the maximal unramified abelian extension of K in which the infinite places split completely, i.e., the maximal abelian subextension of K2IK, we see that G(K2IK1) is the commutator subgroup of G(K2IK). The proof of the principal ideal theorem is thus reduced to the following purely group-theoretic result. 0

(7.6) Theorem. Let G be a finitely generated group, G' its commutator subgroup, and G" the commutator subgroup of G'. If (G : G') < 00, then the transfer

Ver: G /G' ---+ G' /G" is the trivial homomorphism.

We give a proof of this theorem which is due to ERNST WJ7T [141]. In the group ring Z[G] = {LO'EG nO'u I nO' E Z}, we consider the augmentation ideal I G, which is by definition the kernel of the ring homomorphism

Z[G] ---+ Z, L nO'u 1---+ L nO' .

For every subgroup H of G, we have IH ~ IG, and {r - 11 r E H, r i= I} is a Z -basis of I H. We first establish the following lemma, which also has independent interest in that it gives an additive interpretation of the transfer.


(7.7) Lemma. For every subgroup H of tinite index in G, one has a commutative diagram

GIG' Ver

-------+) HI H'

where the homomorphisms 8 are induced by tJ f-+ 8tJ = tJ - 1, and the homomorphism S is given by

Sex mod 11;) = x L: p mod IGIH, pER

for a system of representatives of the left cosets R 3 1 of G I H .

Proof: We first show that the homomorphism

induced by r f-+ 8r = r - 1 has an inverse. The elements p8r, r E H, r =F 1, pER, form a Z -basis of I H + I G I H. Indeed, it follows from

p8r = 8r + 8p8r

that they generate I H + I G I H , and if

0= L: np,TP8r = L: np,T(pr - p) = L: np,Tpr - L:(L:np,T)p, p, r p, r p, r p T

then we conclude that n p , r = 0 because the pr, p are pairwise distinct. Mapping p8r to r mod H', we now have a surjective homomorphism

IH +IGIH ~ HIH'.

It sends 8(pr')8r E IGIH to r'rr,-Ir-I == 1 mod H' because 8(pr')8r = p8(r'r) - p8r' - 8r. It thus induces a homomorphism which is inverse to

(*). In particular, if H = G, we obtain the isomorphism GIG' ~ IGIIl;.

The transfer is now obtained as

Ver(tJ mod G') = TI tJp mod H', pER

where tJ p E H is defined by tJ P = p'tJ p, p' E R. Ver thus induces the homomorphism


given by S(8a mod I~) = LpeR 8ap mod IGIH. From ap = p'ap follows the identity

8p + (8a)p = 8ap + 8p' + 8p'8ap .

Since p' runs through the set R if p does, we get as claimed

S(8p mod I~) == L 8ap == L (8a)p == 8a L p mod IGIH. D peR peR peR

Proof of theorem (7.6): Replacing G by GIG", we may assume that G" = {I}, i.e., that G' is abelian. Let R 3 1 be a system of representatives of left cosets of GIG', and let aI, ... , an be generators of G. Mapping ej = (0, ... ,0,1,0, ... ,0) E Zn to aj, we get an exact sequence

where f is given by an n x n-matrix (mjk) with det(mik) = (G : G'). Consequently,

n n ajmik T:k = 1 with T:k E G' . i=1

The formulae 8(xy) = 8x+8y+8x8y, 8(x-1) = -(8x)x-1 yield by iteration that

n n 8( n aimikT:k) = L(8ai)JLik = 0,

i=1 i=1

where JLik == mik mod IG. In fact, the T:k are products of commutators of the ai and ai-I. We view (JLik) as a matrix over the commutative ring

Z[GIG'] ~ Z[G]/Z[G]IG"

which gives a meaning to the determinant JL = det(JLik) E Z[G I G']. Let ()..kj) be the adjoint matrix of (JLik). Then

(8aj)/-L = L(8ai)JLikAkj == ° mod IGZ[G]IG', i,k

so that (8a)JL == ° mod IGZ[G]IG' = IGIG' for all a. This yields

JL == L p mod Z[G]IG" peR

For if we put JL = LpeR npp, where p = p mod G', then for all U E GIG',

UJL = Lnpup = Lnpp. p p


This implies that all np are equal, hence IL == m LPER p mod Z[G]la', and as

IL == det(mik) == (G : G') == m(G : G') mod la,

we even have m = 1. Applying now lemma (7.7), we see that the transfer is the trivial homomorphism since

S(8a mod l~) == 8a L P == (8a)1L == 0 mod lala'. 0 pER

A problem which is closely related to the principal ideal theorem and which was first put forward by PHIUPP FURTWANaLER is the problem of the class field tower. This is the question whether the class field tower

K = Ko ~ Kl ~ K2 ~ K3 ~ ... ,

where Kj+l is the Hilbert class field of Ki, stops after a finite number of steps. A positive answer would have the implication that the last field in the tower had class number 1 so that in it not only the ideals of K, but in fact all its ideals become principal. This perspective naturally generated the greatest interest. But the problem, after withstanding for a long time all attempts to solve it, was finally decided in the negative by the Russian mathematicians E.S. GOWD and I.R. SAFAREVlt in 1964 (see [48], [24]).

Exercise 1. The decomposition law for the prime ideals p which are ramified in an abelian extension L I K can be formulated like this. Let f be the conductor of L I K , H f ~ J 1 the ideal group for L, and H p the smallest ideal group containing H f of conductor prime to p.

If e = (Hp : Hf) and pi is the smallest power of p which belongs to Hp, then

p = (IlJI ... IlJr)' ,

where the llJi are of degree f over K, and r = if, n = [L : K].

Hint: The class field for H p is the inertia field above p.

The following exercises 2-6 concern a non-abelian example of E. ARTIN.

Exercise 2. The polynomial f (X) = X5 - X + 1 is irreducible. The discriminant of a root a (i.e., the discriminant of Z[a]) is d = 19· 151.

Hint: The discriminant of a root of X5 + aX + b is 55b4 + 28a5.

Exercise 3. Let k = Q(a). Then Z[a] is the ring Ok of integers of k.

Hint: The discriminant of Z[a] equals the discriminant of Ok because on the one hand, both differ only by a square, and on the other hand, it is squarefree. The transition matrix from 1, a, .•• , an-I to an integral basis WI, ••• , Wn of Ok is therefore invertible over Z.


Exercise 4. The decomposition field K 1<Ql of I(X) has as Galois group the symmetric group 6 5 , i.e., it is of degree 120.

Exercise S. K has class number 1.

Hint: Show, using chap. I, §6, exercise 3, that every ideal class of K contains an ideal a with m( a) < 4. If m( a) i= l, then a has to be a prime ideal p such that m(p) = 2 or 3. Hence ok/p = Z/2Z or = Z/3Z, so I has a root mod 2 or 3, which is not the case.

Exercise 6. Show that K I <Ql(JI9. 151) is a (non-abelian!) unramified extension.

Exercise 7. For every Galois extension LIK of finite algebraic number fields, there exist infinitely many finite extensions K' such that L n K' = K, and such that LK'IK' is unramified.

Hint: Let S be the set of places ramified in LIK, and let Lp = Kp(ap). By the approximation theorem, choose an algebraic number a which, for every PES, is close to ap when embedded into Kp. Then Kp(ap) S; Kp(a) by Krasner's lemma, chap. II, §6, exercise 2. Put K' = K(a) and show that LK'IK' is unramified. To show that a can be chosen such that L n K' = K use (3.7), and the fact that G(L I K) is generated by elements of prime power order.

§ 8. The Reciprocity Law of the Power Residues

In class field theory Gauss's reciprocity law meets its most general and definite formulation. Let n be a positive integer :::: 2 and K a number field containing the group /Ln of n -th roots of unity. In chap. V, § 3, we introduced, for every place p of K, the n -th Hilbert symbol

( -t-) : K; x K; --+ /Ln·

It is given via the norm residue symbol by

These symbols all fit together in the following product formula.

(8.1) Theorem. For a, b E K* one has

§ 8. The Reciprocity Law of the Power Residues 415

Proof: From (5.7), we find

and hence the theorem. D

In chap. V, § 3, we defined the n -th power residue symbol in terms of the Hilbert symbol:

(~) = CT;a), where p is a prime ideal of K not dividing n, a E Up, and rr is a prime element of K p. We have seen that this definition does not depend on the choice of the prime element rr and that one has

(~) = 1 {:=} a == an mod p,

and more generally

(~) == a(q-l)/n mod p, q = l)1(p).

(8.2) Definition. For every ideal b = Dpfn p Vp prime to n, and every number a prime to b, we define the n-th power residue symbol by

(a)vp Here p = 1 when vp = O.

The power residue symbol (~) is obviously multiplicative in both

arguments. If b is a principal ideal (b), we write for short (~) = (~). We now prove the general reciprocity law for the n-th power residues.

(8.3) Theorem. If a, bE K* are prime to each other and to n, then


Proof: If P is prime to bnoo, then we have

where Jr is a prime element of Kp. For if we put a = uJrvp(a), then (U,/) = 1 because u, b E Up. For the same reason, we find

( a~b) __ 1 t' for p prime to abnoo.

(8.1) then gives

Here pi (b) means that p occurs in the prime decomposition of (b). 0

Gauss's reciprocity law, for which we gave an elementary proof using the theory of Gauss sums in chap. I, (8.6), in the case of two odd prime numbers p, I, is contained in the general reciprocity law (8.3) as a special case. For if we substitute, in the case K = Q, n = 2, into formula (8.3) the explicit description (chap. V, (3.6» of the Hilbert symbol (apb) for p = 2 and p = 00, we obtain the following theorem, which is more general than chap. I, (8.6).

(8.4) Gauss's Reciprocity Law. Let K = Q, n = 2, and let a and b be odd, relatively prime integers. Then one has

(a) (b) a-I b-l sgna-l sgnb-l b -;; = (-1) 2 2 (-1)----z- 2 ,

and for positive odd integers b, we have the two "supplementary theorems"

(-1) b-l b = (-l)"T, (2) b2_1 b =(-1)-8-.

For the last equation we need again the product formula:

§ 8. The Reciprocity Law of the Power Residues 417

The symbol (~) is called the Jacobi symbol, or also the quadratic residue symbol (although, for b not a prime number, the condition that the symbol (~) = 1 is no longer equivalent to the condition that a is a quadratic residue modulo b).

In the above formulation, the reciprocity law allows us to compute simply by iteration the quadratic residue symbol (~) , as is shown in the following example:

( 40077) (65537) (25460) (22) ( 6365) (40077) 65537 = 40077 = 40077 = 40077 40077 = 6365 =

( 1887) (6365) (704 ) (43 ) ( 11) ( 1887) 6365 = 1887 = 1887 = 1887 1887 = - U =

-( :1) = -( ~) (:1) = (:1) = -C31 ) = -(~) = 1.

Class field theory originated from Gauss's reciprocity law. The quest for a similar law for the n -th power residues dominated number theory for a long time, and the all-embracing answer was finally found in Artin's reciprocity law. The above reciprocity law (8.3) of the power residues now appears as a simple and special consequence of Artin's reciprocity law. But to really settle the original problem, class field theory was still lacking the

explicit computation of the Hilbert symbols (a;,b) for plnoo. This was

finally completed in the 1960s by the mathematician HELMUT BRUCKNER, see chap. V, (3.7).

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