On the Stickelberger Ideal and the Relative Class Number

Abstract

Let $k$ be any imaginary abelian field, $R$ the integral group ring of $G = {\text {Gal}}(k/\mathbb {Q})$, and $S$ the Stickelberger ideal of $k$. Roughly speaking, the relative class number ${h^ - }$ of $k$ 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 ${c^ - }$ in $\frac {1} {2}\mathbb {N} = \{ n/2;n \in \mathbb {N}\}$, which can be described without ${h^ - }$ and is of lower than ${h^ - }$ if the conductor of $k$ is sufficiently large (cf. [6, 9, 10]; see also [5]). We shall prove that $2{c^ - }$, a natural number, divides $2{([k:\mathbb {Q}]/2)^{[k:\mathbb {Q}]/2}}$. In particular, if $k$ varies through a sequence of imaginary abelian fields of degrees bounded, then ${c^ - }$ takes only a finite number of values. On the other hand, it will be shown that ${c^ - }$ can take any value in $\frac {1} {2}\mathbb {N}$ when $k$ ranges over all imaginary abelian fields. In this connection, we shall also make a simple remark on the divisibility for the relative class number of cyclotomic fields.