On the Stickelberger ideal and the relative class number

Abstract

Let k be any imaginary abelian fieldX R the integral group ring of G = Gal(k/(2) and S the Stickelberger ideal of k. Roughly speakingX the relative class number hof k is expressed as the index of S in a certain ideal A of R described by means of G and the complex conjugation of k; c-h= [A: S] with a rational number cin -NJ = {n/2;n e NJ} which can be described without hand is of lower than hif the conductor of k is sufficiently large (cf. [6 9 1O]; see also [5]). We shall prove that 2ca natural numberX divides 2([k: (2]/2)lk 1/2. In particularX if k varies through a sequence of imaginary abelian fields of degrees boundedX then ctakes only a finite number of values. On the other handX it will be shown that ccan take any value in 2NJ when k ranges over all imaginary abelian fields. In this connectionX we shall also make a simple remark on the divisibility for the relative class number of cyclotomic fields. Let 2, C;!!, 1R, and C denote the rational integer ring, the rational number field, the real number field, and the complex number field, respectively. A finite abelian extension over C;!! contained in C will be called an abelian field. Let k be an imagi- nary abelian field, namely, an abelian field not contained in 1R. We denote by R(k) the group ring of the Galois group G = Gal(k/C;!!) over z and by s(H), for any subgroup H of G, the sum in R(k) of all elements in H. Put A(k) = { E R(k); (1 + jk)°l = as(G) for some a E E}, where ik denotes the complex conjugation of k. Let hk denote the relative class number of k (i.e., the so-called first factor of the class number of k), Qk the unit index of k, 9k the number of distinct prime numbers ramified in k, and S(k) the Stickelberger ideal of k in the sense of Iwasawa-Sinnott, which is an additive sub- group of A(k) with finite index (for the definition of the Stickelberger ideal, see [6, 10]). We define ck as the ratio of the index [A(k): S(k)] to hk: Ck hk = [A(k): S(k)] The product QkCk is known to be a natural number and is determined by Sinnott in various cases, for example, in the case 9k = 1 or 2 (cf. [10]). He has also shown in [9] that, if k is a cyclotomic field, then Ck = 2b where b = 0 or 29k-1-1 according as 9k = 1 or 9k > 2 (for the case 9k = 1, see [6]). In this paper, we shall give an additional result concerning the range of Ck . Received by the editors July 31 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary llR20 llR29; Secondary llN25 llR18.