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

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 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(O(F)G). 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.