On the Galois module structure of ideal class groups

Abstract

Let $K/k$ be a Galois extension of a number field of degree $n$ and $p$ a prime number which does not divide $n$. The study of the $p$-rank of the ideal class group of $K$ by using those of intermediate fields of $K/k$ has been made by Iwasawa, Masley et al., attaining the results obtained under respective constraining assumptions. In the present paper we shall show that we can remove these assumptions, and give more general results under a unified viewpoint. Finally, we shall add a remark on the class numbers of cyclic extensions of prime degree of $\mathbb{Q}$.