On the relative Galois module structure of rings of integers in tame extensions

TL;DR

This paper investigates the structure of rings of integers in tame Galois extensions over number fields, using algebraic K-theory to understand the realizable classes in the locally free class group, especially for groups of odd order.

Contribution

It introduces a new approach using relative algebraic K-theory to analyze realizable classes in Galois module theory, extending classical results to the context of tame extensions.

Findings

Realisable classes form a subgroup when G has odd order under certain conditions.

The approach links Galois module structure with relative algebraic K-theory.

Partial analogue of Shafarevich's theorem for soluble groups in Galois module setting.

Abstract

Let $F$ be a number field with ring of integers $O_F$ and let $G$ be a finite group. We describe an approach to the study of the set of realisable classes in the locally free class group $Cl(O_FG)$ of $O_FG$ that involves applying the work of the second-named author in the context of relative algebraic $K$ theory. When $G$ is of odd order, we show (subject to certain conditions) that the set of realisable classes is a subgroup of $Cl(O_FG)$. This may be viewed as being a partial analogue of a classical theorem of Shafarevich on the inverse Galois problem for soluble groups in the setting of Galois module theory.