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

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 $Cl(O_{F}G)$ of $O_{F}G$ 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 $Cl(O_{F}G)$. 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.

Final version. To appear in Algebra and Number Theory

References in corpus (4)

• On the Galois module structure of the square root of the inverse different in abelian extensions
• From Galois module classes to Steinitz classes
• On the realizable classes of the square root of the inverse different in the unitary class group
• On twisted forms and relative algebraic K-theory

Cited by in corpus (1)

• On the square root of the inverse different