ACTA ARITHMETICA XXII.4 (1995) Ideal class groups of cyclotomic number fields I by Franz emmermeyer (Heidelberg) 1. Notation. et K be number fields; we will use the following notation: O K is the ring of integers of K; E K is its group of units; W K is the group of roots of unity contained in K; w K is the order of W K ; Cl(K) is the ideal class group of K; [a] is the ideal class generated by the ideal a; K 1 denotes the Hilbert class field of K, that is the maximal abelian extension of K which is unramified at all places; j K denotes the transfer of ideal classes for number fields K, i.e. the homomorphism Cl(K) Cl() induced by mapping an ideal a to ao ; κ /K denotes the capitulation kernel ker j K ; Now let K be a CM-field, i.e. a totally complex quadratic extension of a totally real number field; the following definitions are standard: σ is complex conjugation; K + denotes the maximal real subfield of K; this is the subfield fixed by σ; Cl (K) is the kernel of the map N K/K + : Cl(K) Cl(K + ) and is called the minus class group; h (K) is the order of Cl (K), the minus class number; Q(K) = (E K : W K E K +) {1, 2} is Hasse s unit index. We will need a well known result from class field theory. Assume that K are CM-fields; then ker(n /K : Cl() Cl(K)) has order ( K 1 : K). Since K/K + is ramified at the infinite places, the norm N K/K + : Cl(K) Cl(K + ) is onto. 1991 Mathematics Subject Classification: Primary 11R18; Secondary 11R29. [347]
348 F. emmermeyer 2. Hasse s unit index. Hasse s book [H] contains numerous theorems (Sätze 14 29) concerning the unit index Q() = (E : W E K ), where K = + is the maximal real subfield of a cyclotomic number field. Hasse considered only abelian number fields /Q, hence he was able to describe these fields in terms of their character groups X(); as we are interested in results on general CM-fields, we have to proceed in a different manner. But first we will collect some of the most elementary properties of Q() (see also [H] and [W]; a reference Satz always refers to Hasse s book [H]) in Proposition 1. et K be CM-fields; then (a) (Satz 14) Q() = (E : W E +) = (E σ 1 : W 2) = (Eσ+1 : E 2 ); + in particular, Q() {1, 2}; (b) (Satz 16, 17) If Q() = 2 then κ / + = 1; (c) (Satz 25) If + contains units with any given signature, then Q() = 1; (d) (Satz 29) Q(K) Q() (W : W K ); (e) (compare Satz 26) Suppose that N /K : W /W 2 W K /W K 2 is onto. Then Q() Q(K); (f) ([HY, emma 2]) If ( : K) is odd, then Q() = Q(K); (g) (Satz 27) If = Q(ζ m ), where m 2 mod 4 is composite, then Q() = 2; (h) (see Example 4 below) et K 1 Q(ζ m ) and K 2 Q(ζ n ) be abelian CM-fields, where m = p µ and n = q ν are prime powers such that p q, and let K = K 1 K 2 ; then Q(K) = 2. The proofs are straightforward: (a) The map ε ε σ 1 induces an epimorphism E E σ 1 /W 2. If ε σ 1 = ζ 2 for some ζ W, then (ζε) σ 1 = 1, and ζε E +. This shows that σ 1 gives rise to an isomorphism E /W E + E σ 1 /W 2, hence we have (E : W E +) = (E σ 1 : W 2 ). The other claim is proved similarly. (b) Since W /W 2 is cyclic of order 2, the first claim follows immediately from (a). Now let a be an ideal in O K such that ao = αo. Then α σ 1 = ζ for some root of unity ζ, and Q() = 2 shows that ζ = ε σ 1 for some ε E. Now αε 1 generates a and is fixed by σ, hence lies in K. This shows that a is principal in K, i.e. that κ / + = 1. (c) Units in + which are norms from are totally positive; our assumption implies that totally positive units are squares, hence we get E σ+1 = E 2, and our claim follows from (a). + (d) First note that (W : W K ) = (W 2 : W K 2 ); then
Q() (W : W K ) = (E σ 1 = (E σ 1 Ideal class groups 349 : W 2 )(W 2 : W 2 K) = (E σ 1 : E σ 1 K ) Q(K) : E σ 1 K )(Eσ 1 K : W K) 2 proves the claim. (e) Since Q() = 2, there is a unit ε E such that ε σ 1 = ζ generates W /W 2. Taking the norm to K shows that (N /Kε) σ 1 = N /K (ζ) generates W K /WK 2, i.e. we have Q(K) = 2. (f) If ( : K) is odd, then (W : W K ) is odd, too, and we get Q(K) Q() from (d) and Q() Q(K) from (e). (g) In this case, 1 ζ m is a unit, and we find (1 ζ m ) 1 σ = ζ m. Since ζ m W \ W 2, we must have Q() = 2. (h) First assume that m and n are odd. A subfield F = Q(ζ m ), where m = p µ is an odd prime power, is a CM-field if and only if it contains the maximal 2-extension contained in, i.e. if and only if ( : F ) is odd. Since (Q(ζ m ) : K 1 ) and (Q(ζ n ) : K 2 ) are both odd, so is (Q(ζ mn ) : K 1 K 2 ); moreover, Q(ζ mn ) has unit index Q = 2, hence the assertion follows from (f) and (g). Now assume that p = 2. If 1 K 1, then we must have K 1 = Q(ζ m ) for m = 2 α and some α 2 (complex subfields of the field of 2 µ th roots of unity containing 1 necessarily have this form). Now n is odd and K 2 Q(ζ n ) is complex, hence (Q(ζ n ) : K 2 ) is odd. By (f) it suffices to show that K 1 (ζ n ) = Q(ζ mn ) has unit index 2, and this follows from (g). If 1 K 1, let K 1 = K 1 (i); then K 1 = Q(ζ m ) for m = 2 α and some α 2, and in the last paragraph we have seen that Q( K 1 K 2 ) = 2. Hence we only need to show that the norm map N : W K1 /W 2 K 1 W K1 /W 2 K 1 is onto: since (W K1 : W K 2 K1 ) is odd, this implies 2 = Q( K 1 K 2 ) Q(K 1 K 2 ) by (e). But the observation that the non-trivial automorphism of Q(ζ m )/K 1 maps ζ m to ζm 1 implies at once that N(ζ m ) = 1, and 1 generates W K1 /WK 2 1. Now let be a CM-field with maximal real subfield K; we will call /K essentially ramified if = K( α ) and there is a prime ideal p in O K such that the exact power of p dividing αo K is odd; it is easily seen that this does not depend on which α we choose. Moreover, every ramified prime ideal p above an odd prime p is necessarily essentially ramified. We leave it as an exercise to the reader to verify that our definition of essential ramification coincides with Hasse s [H, Sect. 22]; the key observation is the ideal equation (4α) = a 2 d, where d = disc(k( α )/K) and a is an integral ideal in O K. We will also need certain totally real elements of norm 2 in the field of
350 F. emmermeyer 2 m th roots of unity: to this end we define π 2 = 2 = 2 + ζ 4 + ζ 1 4, π 3 = 2 + 2 = 2 + ζ 8 + ζ 1 8,.. π n = 2 + π n 1 = 2 + ζ 2 n + ζ 1 2 n. et m 2, = Q(ζ 2 m+1) and K = Q(π m ); then /K is an extension of type (2, 2) with subfields K 1 = Q(ζ 2 m), K 2 = Q( π m ) and K 3 = Q( π m ). Moreover, K 2 /K and K 3 /K are essentially ramified, whereas K 1 /K is not. Theorem 1. et be a CM-field with maximal real subfield K. (i) If w 2 mod 4, then: 1. If /K is essentially ramified, then Q() = 1, and κ /K = 1. 2. If /K is not essentially ramified, then = K( α ) for some α O K such that αo K = a 2, where a is an integral ideal in O K, and (a) Q() = 2 if a is principal, and (b) Q() = 1 and κ /K = [a] if a is not principal. (ii) If w 2 m mod 2 m+1, where m 2, then /K is not essentially ramified, and: 1. If π m O K is not an ideal square, then Q() = 1 and κ /K = 1. 2. If π m O K = b 2 for some integral ideal b, then (a) Q() = 2 if b is principal, and (b) Q() = 1 and κ /K = [b] if b is not principal. For the proof of Theorem 1 we will need the following emma 1. et = K( π ), and let σ denote the non-trivial automorphism of /K. Moreover, let b be an ideal in O K such that bo = (β) and β σ 1 = 1 for some β. Then πo K is an ideal square in O K. If, on the other hand, β σ 1 = ζ, where ζ is a primitive 2 m th root of unity, then π m O K is an ideal square in O K. P r o o f. We have (β π ) σ 1 = 1, hence β π K. Therefore b and c = (β π ) are ideals in O K, and (cb 1 ) 2 = πo K proves our claim. Now assume that β σ 1 = ζ; then σ fixes (1 ζ)β 1, hence ((1 ζ)β) and c = (1 ζ) = c σ are ideals in O K, and c 2 = N /K (1 ζ) = (2 + ζ + ζ 1 )O K is indeed an ideal square in O K as claimed. P r o o f o f T h e o r e m 1. (i) Assume that w 2 mod 4.
Ideal class groups 351 C a s e 1: /K is essentially ramified. Assume we had Q() = 2; then E σ 1 = W, hence there is a unit ε E such that ε σ 1 = 1. Write = K( π ), and apply emma 1 to b = (1), β = ε: this will yield the contradiction that /K is not essentially ramified. C a s e 2: /K is not essentially ramified. Then = K( α ) for some α O K such that αo K = a 2, where a is an integral ideal in O K. (a) If a is principal, say a = βo K, then there is a unit ε E K such that α = β 2 ε, and we see that = K( ε ). Now ε σ 1 = 1 is no square since w 2 mod 4, and Proposition 1(a) gives Q() = 2. (b) If a is not principal, then the ideal class [a] capitulates in /K because ao = αo. Proposition 1(b) shows that Q() = 1. (ii) Assume that w 2 m mod 2 m+1 for some m 2. C a s e 1: Assume that Q() = 2 or κ /K 1. Then emma 1 says that π m O K = b 2 is an ideal square in O K contrary to our assumption. C a s e 2: π m = b 2 is an ideal square in O K. If b is not principal, then bo = (1 ζ) shows that κ /K = [b], and Proposition 1(b) gives Q() = 1. If, on the other hand, b = βo K, then ηβ 2 = π m for some unit η E K. If η were a square in O K, then π m would also be a square, and = K( 1 ) would contain the 2 m+1 th roots of unity. Now ηβ 2 = π m = ζ 1 (1 + ζ) 2, hence ηζ is a square in, and we have Q() = 2 as claimed. R e m a r k. For /Q abelian, Theorem 1 is equivalent to Hasse s Satz 22; we will again only sketch the proof: suppose that w 2 m mod 2 m+1 for some m 2, and define = (ζ 2 m+1), K = R. Then K /K is essentially ramified if and only if π m is not an ideal square in O K (because K = K(π m+1 ) = K( π m )). The asserted equivalence should now be clear. Except for the results on capitulation, Theorem 1 is also contained in [O] (for general CM-fields). Examples. 1. Complex subfields of Q(ζ p m), where p is prime, have unit index Q() = 1 (Hasse s Satz 23) and κ / + = 1: since p ramifies completely in Q(ζ p m)/q, / + is essentially ramified if p 2, and the claim follows from Theorem 1. If p = 2 and / + is not essentially ramified, then we must have = Q(ζ 2 µ) for some µ N, and we find Q() = 1 by Theorem 1(ii.1). 2. = Q(ζ m ) has unit index Q() = 1 if and only if m 2 mod 4 is a prime power (Satz 27). This follows from Example 1 and Proposition 1(e). 3. If K is a CM-field, which is essentially ramified at a prime ideal p above p N, and if F is a totally real field such that p disc F, then Q() = 1 and κ / + = 1 for = KF : this is again due to the fact that either / + is essentially ramified at the prime ideals above p, or p = 2 and
352 F. emmermeyer K = K + ( 1 ). In the first case, we have Q() = 1 by Theorem 1(i.1), and in the second case by Theorem 1(ii.1). 4. Suppose that the abelian CM-field K is the compositum K = K 1... K t of fields with pairwise different prime power conductors; then Q(K) = 1 if and only if exactly one of the K i is imaginary (Uchida [U, Prop. 3]). The proof is easy: if there is exactly one complex field among the K j, then Q(K) = 1 by Example 3. Now suppose that K 1 and K 2 are imaginary; we know Q(K 1 K 2 ) = 2 (Proposition 1(h)), and from the fact that the K j have pairwise different conductors we deduce that (W K : W K1 K 2 ) 1 mod 2. Now the claim follows from Hasse s Satz 29 (Proposition 1(c)). Observe that κ K/K + = 1 in all cases. 5. Cyclic extensions /Q have unit index Q() = 1 (Hasse s Satz 24): et F be the maximal subfield of such that (F : Q) is odd. Then F is totally real, and 2 disc F (this follows from the theorem of Kronecker and Weber). Similarly, let K be the maximal subfield of such that (K : Q) is a 2-power: then K is a CM-field, and = F K. If K/K + is essentially ramified at a prime ideal p above an odd prime p, then so is / +, because /Q is abelian, and all prime ideals in F have odd ramification index. Hence the claim in this case follows by Example 3 above. If, however, K/K + is not essentially ramified at a prime ideal p above an odd prime p, then disc K is a 2-power (recall that K/Q is cyclic of 2-power degree). Applying the theorem of Kronecker and Weber, we find that K Q(ζ), where ζ is some primitive 2 m th root of unity. If K/K + is essentially ramified at a prime ideal above 2, then so is / +, and Theorem 1 gives us Q() = 1. If K/K + is not essentially ramified at a prime ideal above 2, then we must have K = Q(ζ), where ζ is a primitive 2 m th root of unity; but now π m O + is not the square of an integral ideal, and we have Q() = 1 by Theorem 1. Alternatively, we may apply Proposition 1(e) and observe that Q(K) = 1 by Example 1. 6. et p 1 mod 8 be a prime such that the fundamental unit ε 2p of Q( 2p ) has norm +1 (by [S], there are infinitely many such primes; note also that Nε 2p = +1 (2, 2p ) is principal). Put K = Q(i, 2p ) and = Q(i, 2, p ). Then Q(K) = 2 by Theorem 1(ii.2)(a), whereas the fact that is the compositum of Q(ζ 8 ) and Q( p ) shows that Q() = 1 (Example 4). This generalization of enstra s example given by Martinet in [H] is contained in Theorem 4 of [HY], where several other results of this kind can be found. 3. Masley s theorem h m h mn. Now we can prove a theorem which will contain Masley s result h (K) h () for cyclotomic fields K = Q(ζ m ) and = Q(ζ mn ) as a special case:
Theorem 2. et K be CM-fields; then h (K) h () κ / + and the last quotient is a power of 2. Ideal class groups 353 ( K 1 : K) ( + (K + ) 1 : K + ), P r o o f. et ν K and ν denote the norms N K/K + and N / +, respectively; then the following diagram is exact and commutative: 1 Cl () Cl() ν Cl( + ) 1 N N N + 1 Cl (K) Cl(K) The snake lemma gives us an exact sequence ν K Cl(K + ) 1 1 ker N ker N ker N + cok N cok N cok N + 1. et h(/k) denote the order of ker N, and let h (/K) and h( + /K + ) be defined accordingly. The remark at the end of Section 1 shows cok N = ( K 1 : K), cok N + = ( + (K + ) 1 : K + ). The alternating product of the orders of the groups in exact sequences equals 1, so the above sequence implies h (/K) h( + /K + ) cok N = h(/k) cok N cok N +. The exact sequence 1 ker N Cl () Cl (K) cok N 1 gives us h (/K) h (K) = h () cok N. Collecting everything we find that ( ) h (K) h(/k) h( + /K + ) (+ (K + ) 1 : K + ) ( K 1 = h (). : K) Now the claimed divisibility property follows if we can prove that h( + /K + ) divides h(/k) κ / +. But this is easy: exactly h( + /K + )/ κ / + ideal classes of ker N + Cl( + ) survive the transfer to Cl(), and if the norm of + /K + kills an ideal class c Cl( + ), the same thing happens to the transferred class c j when the norm of /K is applied. We remark in passing that κ / + 2 (see Hasse [H, Satz 18]). It remains to show that ( K 1 : K)/( + (K + ) 1 : K + ) is a power of 2. Using induction on ( : K), we see that it suffices to prove that if /K is an unramified abelian extension of CM-fields of odd prime degree ( : K) = q, then so is + /K +. Suppose otherwise; then there exists a
354 F. emmermeyer finite prime p which ramifies, and since + /K + is cyclic, p has ramification index q. Now /K + is cyclic of order 2q, hence K must be the inertia field of p, contradicting the assumption that /K is unramified. We conclude that + /K + is also unramified, and so odd factors of ( K 1 : K) cancel against the corresponding factors of ( + (K + ) 1 : K + ). Corollary 1 ([OO]). et K be CM-fields such that ( : K) is odd; then h K h (). P r o o f. From ( ) and the fact that ( + (K + ) 1 : K + ) = 1 (this index is a power of 2 and divides ( : K), which is odd), we see that it is sufficient to show that h( + /K + ) h(/k). This in turn follows if we can prove that no ideal class from ker N + Cl( + ) capitulates when transferred to Cl(). Assume therefore that κ / + = [a]. If w 2 mod 4, then by Theorem 1(i.2) we may assume that = + ( α ), where αo + = a 2. Since ( : K) is odd, we can choose α O K +, hence N + (a) = a (:K) shows that the ideal class [a] is not contained in ker N +. The proof in the case w 0 mod 4 is completely analogous. R e m a r k. For any prime p, let Cl p (K) denote the p-sylow subgroup of Cl (K); then Cl p (K) Cl p () for every p ( : K). This is trivial, because ideal classes with order prime to ( : K) cannot capitulate in /K. Corollary 2 ([MM]). If K = Q(ζ m ) and = Q(ζ mn ) for some m, n N, then h (K) h (). P r o o f. We have shown in Section 2 that j K + K and j + are injective in this case. Moreover, /K does not contain a non-trivial subfield of K 1 (note that p is completely ramified in /K if n = p, and use induction). The special case m = p a, n = p of Corollary 2 can already be found in [We]. Examples of CM-fields /K such that h (K) h () have been given by Hasse [H]; here are some more: 1. et d 1 { 4, 8, q (q 3 mod 4)} be a prime discriminant, and suppose that d 2 > 0 is the discriminant of a real quadratic number field such that (d 1, d 2 ) = 1. Put K = Q( d 1 d 2 ) and = Q( d 1, d 2 ); then Q() = 1 and κ / + = 1 by Example 4, and ( K 1 : K) = 2 ( + (K + ) 1 : K + ) since /K is unramified but + /K + is not. The class number formula (1) below shows that in fact h (K) h (). 2. et d 1 = 4, d 2 = 8m for some odd m N, and suppose that 2 = (2, 2m ) is not principal in O k, where k = Q( 2m ). Then h (K) h () for K = Q( 2m ), = Q( 1, 2m ). Here ( K 1 : K) = ( + (K + ) 1 : K + ), but κ / + = [2], since 2O = (1 + i). This example shows that we cannot drop the factor κ / + in Theorem 2.
Ideal class groups 355 Other examples can be found by replacing d 1 in Example 2 by d 1 = 8 or d 2 by d 2 = 4m, m N odd. The proof that in fact h (K) h () for these fields uses Theorem 1, as well as Propositions 2 and 3 below. 4. Metsänkylä s factorization. An extension /K is called a V 4 - extension of CM-fields if 1. /K is normal and Gal(/K) V 4 = (2, 2); 2. Exactly two of the three quadratic subfields are CM-fields; call them K 1 and K 2, respectively. This implies, in particular, that K is totally real, and that is a CM-field with maximal real subfield + = K 3. We will write Q 1 = Q(K 1 ), W 1 = W K1, etc. ouboutin [ou, Prop. 13] has given an analytic proof of the following class number formula for V 4 -extension of CM-fields, which contains emma 8 of Ferrero [F] as a special case: Proposition 2. et /K be a V 4 -extension of CM-fields; then h () = Q() w h (K 1 )h (K 2 ). Q 1 Q 2 w 1 w 2 P r o o f. Kuroda s class number formula (for an algebraic proof see []) yields (1) h() = 2 d κ 2 υ q()h(k 1 )h(k 2 )h( + )/h(k) 2, where d = (K : Q) is the number of infinite primes of K ramified in /K; κ = d 1 is the Z-rank of the unit group of K; υ = 1 if and only if all three quadratic subfields of /K can be written as K( ε ) for units ε E K, and υ = 0 otherwise; q() = (E : E 1 E 2 E 3 ) is the unit index for extensions of type (2, 2); here E j is the unit group of K j (similarly, let W j denote the group of roots unity in j ). Now we need to find a relation between the unit indices involved; we assert Proposition 3. If /K is a V 4 -extension of CM-fields, then Q() w = 2 1 υ q(). Q 1 Q 2 w 1 w 2 P r o o f o f P r o p o s i t i o n 3. We start with the observation Q() = (E : W E 3 ) = (E : E 1 E 2 E 3 ) (E 1E 2 E 3 : W 1 W 2 E 3 ) (W E 3 : W 1 W 2 E 3 ).
356 F. emmermeyer In [] we have defined groups E j = {ε E j : N j ε is a square in E K }, where N j denotes the norm of K j /K; we have also shown that (E 1 E 2 E 3 : E 1E 2E 3) = 2 υ (E j : E j ) and E j /Ej E K/N j E j. Now Proposition 1(a) gives (E K : N j E j ) = Q j for j = 1, 2, and we claim w 1. (W E 3 : W 1 W 2 E 3 ) = (W : W 1 W 2 ) = 2 ; w 1 w 2. E 2 1E2E 3 = W 1 W 2 E3; 3. (W 1 W 2 E 3 : W 1 W 2 E3) = (E 3 : E3). This will give us (2) Q() = 2 1 υ q()q 1 Q 2 w 1 w 2 w, completing the proof of Proposition 3; inserting (2) into equation (1) and recalling the definition of the minus class number yields ouboutin s formula. We still have to prove the three claims above: 1. W E 3 /W 1 W 2 E 3 W /(W W 1 W 2 E 3 ) W /W 1 W 2, and the claim follows from W 1 W 2 = { 1, +1}; 2. We only need to show that E 1E 2E 3 W 1 W 2 E 3; but Proposition 1(a) shows that ε E 1 ε σ+1 E 2 K ε W 1E K, and this implies the claim; 3. W 1 W 2 E 3 /W 1 W 2 E 3 E 3 /E 3 W 1 W 2 E 3 E 3 /E 3. Combining the result of Section 3 with Proposition 2, we get the following Theorem 3. et 1 and 2 be CM-fields, and let = 1 2 and K = + 1 + 2 ; then /K is a V 4-extension of CM-fields with subfields K 1 = 1 + 2, K 2 = + 1 2, K 3 = +, and h () = Q() w h ( 1 )h ( 2 )T 1 T 2, Q 1 Q 2 w 1 w 2 where T 1 = h ( 1 + 2 )/h ( 1 ) and T 2 = h ( 2 + 1 )/h ( 2 ). If we assume that κ 1 = κ 2 = 1 (κ 1 is the group of ideal classes capitulating in 1 + 2 /K and κ 2 is defined similarly) and that then T 1 and T 2 are integers. ( 1 + 2 1 1 : 1 ) = ( + 1 + 2 (+ 1 )1 : + 1 ), ( 2 + 1 1 2 : 2 ) = ( + 2 + 1 (+ 2 )1 : + 2 ), P r o o f. Theorem 3 follows directly from Theorem 2 and Proposition 2.
Ideal class groups 357 The following Hasse diagram explains the situation: K 1 + K 2 K 2 1 + 2 @ @@ @ @@ + 1 Q { {{ { {{ C CCCC C B BB B BB C CC { {{ C CC { {{ } } B BBBB B Now let m = p µ and n = q ν be prime powers, and suppose that p q. Moreover, let 1 Q(ζ m ) and 2 Q(ζ n ) be CM-fields. Then (1) Q() = 2, Q 1 = Q( 1 + 2 ) = Q 2 = Q( 2 + 1 ) = 1: this has been proved in Proposition 1(h) and Example 4 in Section 2; (2) w 1 w 2 = 2w (obviously); (3) κ 1 = κ 2 = 1: see Example 4 in Section 2; (4) ( 1 + 2 1 1 : 1 ) = ( + 1 + 2 (+ 1 )1 : + 1 ): this, as well as the corresponding property for K 2, is obvious, because the prime ideals above p and q ramify completely in / 2 and / 1, respectively. In particular, we have the following Corollary ([M]). et 1 Q(ζ m ) and 2 Q(ζ n ) be CM-fields, where m = p µ and n = q ν are prime powers, and let = 1 2 ; then } } h () = h ( 1 )h ( 2 )T 1 T 2, where T 1 = h ( 1 + 2 )/h ( 1 ) and T 2 = h ( 2 + 1 )/h ( 2 ) are integers. It still remains to identify the character sums T 01 and T 10 in [M] with the class number factors T 1 and T 2 given above. But this is easy: the character group X( 1 ) corresponding to the field 1 is generated by a character χ 1, and it is easily seen that B BB B BB
358 F. emmermeyer X( 1 ) = χ 1, X( 1 + 2 ) = χ 1, χ 2 2, X( 2 ) = χ 2, X( 2 + 1 ) = χ 2, χ 2 1, X() = χ 1, χ 2, X( + ) = χ 1 χ 2, χ 2 1. The analytical class number formula for an abelian CM-field K reads (3) h 1 (K) = Q(K)w K ( χ(a)a), 2f(χ) χ X (K) a mod + f(χ) where a mod + f(χ) indicates that the sum is extended over all 1 a f(χ) such that (a, f(χ)) = 1, and X () = X() \ X( + ) is the set of χ X() such that χ( 1) = 1. Applying formula (3) to the CM-fields listed above and noting that Q() = 2, Q( 1 ) = Q( 2 ) = Q( 1 + 2 ) = Q( 2 + 1 ) = 1 and 2w = w 1 w 2, we find h () = h ( 1 ) h 1 ( 2 ) ( χ(a)a), 2f(χ) χ X () a mod + f(χ) where X () is the subset of all χ X () not lying in X ( 1 ) or X ( 2 ). Now define X 1 () = {χ = χ x 1χ y 2 X () : x 1 mod 2, y 0 mod 2}, and let X 2 () be defined accordingly. Then X () = X 1 () X 2 (), and h 1 ( 1 ) ( χ(a)a) = h ( 1 + 2 2f(χ) ), χ X 1 () and we have shown that T 1 = χ X 1 () a mod + f(χ) 1 2f(χ) a mod + f(χ) ( χ(a)a). Comparing with the definition of Metsänkylä s factor T 10, this shows that indeed T 1 = T 10. Acknowldegements. I would like to thank Stéphane ouboutin and Ryotaro Okazaki for several helpful suggestions and for calling my attention to the papers of Horie and Uchida. References [F] B. Ferrero, The cyclotomic Z 2 -extension of imaginary quadratic number fields, Amer. J. Math. 102 (1980), 447 459. [H] H. Hasse, Über die Klassenzahl abelscher Zahlkörper, Springer, Berlin, 1985. [HY] M. Hirabayashi and K. Yoshino, Remarks on unit indices of imaginary abelian number fields, Manuscripta Math. 60 (1988), 423 436. [Ho] K. Horie, On a ratio between relative class numbers, Math. Z. 211 (1992), 505 521.
Ideal class groups 359 [] F. emmermeyer, Kuroda s class number formula, Acta Arith. 66 (1994), 245 260. [ou] S. ouboutin, Determination of all quaternion octic CM-fields with class number 2, J. ondon Math. Soc., to appear. [OO] S. ouboutin, R. Okazaki and M. Olivier, The class number one problem for some non-abelian normal CM-fields, preprint, 1994. [M] T. Metsänkylä, Über den ersten Faktor der Klassenzahl des Kreiskörpers, Ann. Acad. Sci. Fenn. Ser. A I 416, 1967. [MM] J. M. Masley and H.. Montgomery, Cyclotomic fields with unique factorization, J. Reine Angew. Math. 286/287 (1976), 248 256. [O] R. Okazaki, On evaluation of -functions over real quadratic fields, J. Math. Kyoto Univ. 31 (1991), 1125 1153. [S] A. Scholz, Über die ösbarkeit der Gleichung t2 Du 2 = 4, Math. Z. 39 (1934), 95 111. [U] K. Uchida, Imaginary quadratic number fields with class number one, Tôhoku Math. J. 24 (1972), 487 499. [W]. Washington, Introduction to Cyclotomic Fields, Springer, New York, 1982. [We] J. Westlund, On the class number of the cyclotomic number field, Trans. Amer. Math. Soc. 4 (1903), 201 212. ERWIN-ROHDE-STR. 19 D-69120 HEIDEBERG, GERMANY E-mail: HB3@IX.URZ.UNI-HEIDEBERG.DE Received on 19.8.1994 (2658)