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The Steinitz class of a number field extension K/k is an ideal class in thering of integers O_k of k, which, together with the degree [K:k] of theextension determines the O_k-module structure of O_K. We call rt(k,G) theclasses which are Steinitz classes of a tamely ramified G-extension of k. Wewill say that those classes are realizable for the group G; it is conjecturedthat the set of realizable classes is always a group. We define A'-groupsinductively, starting by abelian groups and then considering semidirectproducts of A'-groups with abelian groups of relatively prime order and directproducts of two A'-groups. Our main result is that the conjecture aboutrealizable Steinitz classes for tame extensions is true for A'-groups of oddorder; this covers many cases not previously known. Further we use the sametechniques to determine rt(k,D_n) for any odd integer n. In contrast with manyother papers on the subject, we systematically use class field theory (insteadof Kummer theory and cyclotomic descent).

Steinitz classes of tamely ramified Galois extensions of algebraic number fields