On the Galois module structure of ideal class groups

摘要

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