Steinitz classes of tamely ramified Galois extensions of algebraic number fields

The Steinitz class of a number field extension K/k is an ideal class in the ring of integers O(k) of k, which, together with the degree [K :k] of the extension determines the O(k)-module structure of O(K). We call R(t)(k, G) the set of classes which are Steinitz classes of a tamely ramified G-extension of k. We will say that those classes are realizable for the group G; it is conjectured that the set of realizable classes is always a group. We define A'-groups inductively, starting with abelian groups and then considering semidirect products of A'-groups with abelian groups of relatively prime order and direct products of two A'-groups. Our main result is that the conjecture about realizable Steinitz classes for tame extensions is true for A'-groups of odd order; this covers many cases not previously known. Further we use the same techniques to determine R(t)(k,D(n)) for any odd integer n. In contrast with many other papers on the subject, we systematically use class field theory (instead of Kummer theory and cyclotomic descent). (C) 2010 Elsevier Inc. All rights reserved.