Realizable classes and embedding problems

TL;DR

This paper explores the relationship between the classes of rings of integers in tame Galois extensions of number fields and embedding problems, extending classical results to a broader algebraic context.

Contribution

It establishes a connection between the structure of locally free classes of rings of integers and the theory of embedding problems for Galois extensions.

Findings

The collection of classes is related to embedding problems.

Extension of Noether's theorem to broader algebraic structures.

Provides new insights into the structure of Galois module classes.

Abstract

Let $K$ be a number field with ring of integers $\mathcal{O}_K$ and let $G$ be a finite group. Given a $G$-Galois $K$-algebra $K_h$, let $\mathcal{O}_h$ denote its ring of integers. If $K_h/K$ is tame, then a classical theorem of E. Noether implies that $\mathcal{O}_h$ is locally free over $\mathcal{O}_KG$ and hence defines a class in the locally free class group of $\mathcal{O}_KG$. For $G$ abelian, by combining the work of J. Brinkhuis and L. McCulloh, we prove that the structure of the collection of all such classes is related to the study of embedding problems.