On Realizable Galois Module Classes By The Inverse Different

Let k be a number field and O-k its ring of integers. Let Gamma be a finite group. Let M be a maximal O-k-order in the semi-simple algebra k[Gamma] containing O-k[Gamma]: Let Cl(O-k [Gamma] (resp. Cl(M)) be the locally free classgroup of O-k[Gamma] (resp. M). We denote by R(D-1, O-k [Gamma])(resp. R(D-1, M)) the set of classes Gamma in Cl(O-k[Gamma]) resp. Cl(M))) such that there exists a tamely ramified extension N/k, with Galois group isomorphic to Gamma (Gamma-extension) and the class of D-N/k(-1) (resp. M circle times O-k [Gamma]D-N/k(-1)) is equal to c, where D-N/k is the different of N/k and D-N/k(-1) its inverse different. We say that R(D-1, O-k [Gamma] (resp. R(D-1, M) is the set of realizable Galois module classes by the inverse different. In the present article, combining some of our published results, and a result due to A. Frohlich giving a link between the Galois module class of the ring of integers of a tamely ramified C-extension and that of its inverse different, we explicitly determine R(D-1, O-k[Gamma])(resp. R(D-1, M)) for several groups Gamma and show that it is a subgroup of Cl(O-k[Gamma]) (resp. Cl(M)). In addition, we determine the set of the Steinitz classes of D-N/k(-1), N/k runs through the set of tamely ramified Gamma-extension of k, and prove that is a subgroup of Cl(k), also for several groups Gamma.