Classes de Steinitz d'extensions quaternioniennes généralisées de degré 4 p r

Let p be an odd prime number, r a natural number, and H 4 p rthe generalized quaternion group of order 4 p r . Let k be a number field and Cl( k ) its class group. Let R m ( k , H 4 p r) be the subset of Cl( k ) consisting of those classes which are realizable as Steinitz classes of tame Galois extensions of k with Galois group isomorphic to H 4 p r. In this article, we determine R m ( k , H 4 p r) and show that it is a subgroup of Cl( k ). In particular, R m ( k , H 4 p r) is the full group Cl( k ) if p does not divide the class number of k , or if either k contains a primitive p r th root of unity or p is unramified in k .