Determining class groups as Galois modules up to equivalence for some nonabelian extensions

TL;DR

This paper extends the study of Galois module structures of minus class groups from abelian to certain nonabelian metacyclic extensions, providing a complete classification up to an equivalence relation.

Contribution

It introduces a classification of minus class groups as Galois modules for specific nonabelian metacyclic extensions, generalizing previous abelian results.

Findings

Complete description of Galois modules for certain nonabelian extensions

Extension of previous abelian class group results to nonabelian cases

Introduction of an equivalence relation on modules for classification

Abstract

In previous papers, the Galois module structure of minus class groups was studied for abelian CM extensions. In this paper, we discuss some nonabelian cases, focusing on metacyclic extensions. For a certain class of these, we obtain a complete description of the Galois modules that occur as minus class groups, modulo a certain equivalence relation on modules, which was introduced earlier by the same authors.