Realizable classes and embedding problems

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 ${\mathcal{O}}_{K}G$ and hence defines a class in the locally free class group of ${\mathcal{O}}_{K}G$. 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.

Some of the sections were rewritten to improve the exposition; more explanations are added and several typos are now fixed

References in corpus (1)

• On the Galois module structure of the square root of the inverse different in abelian extensions