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 $Cl\left(O_{F}G\right)$ of $O_{F}G$ that involves applying the work of McCulloh in the context of relative algebraic $K$ theory. For a large class of soluble groups $G$, including all groups of odd order, we show (subject to certain mild conditions) that the set of realisable classes is a subgroup of $Cl\left(O_{F}G\right)$. This may be viewed as being a partial analogue in the setting of Galois module theory of a classical theorem of Shafarevich on the inverse Galois problem for soluble groups.