Steinitz classes of Galois extensions with Galois group with nontrivial center

Let k be a number field and Cl(k) its class group. Let Gamma be a finite group and vertical bar Gamma vertical bar its order. Let R(k, Gamma) (resp. R-m(k, Gamma)) be the subset of Cl(k) consisting of those classes which are realizable as Steinitz classes of Galois extensions (resp. tamely ramified Galois extensions) of k with Galois group isomorphic to Gamma. In the present article, we suppose that Gamma is realizable as Galois group over k of a Galois extension (resp. tame Galois extension) - e.g. Gamma solvable - and the center z(Gamma) of Gamma is non-trivial - e.g. Gamma nilpotent non-trivial. For each prime divisor p of the order of Z(Gamma), we define a natural number n(p). We show that if the class number of k is prime to n(p), then R(k, Gamma) (resp. R-m(k, Gamma)) is the full group Cl(k). For instance, this result applies to a nilpotent group Gamma having even order, with n(2) = vertical bar Gamma vertical bar/2. (C) 2012 Elsevier Inc. Tous droits reserves.