Classes de Steinitz dʼextensions galoisiennes à groupe de Galois de centre non trivial
Résumé
Let k be a number field and Cl(k) its class group. Let Γ be a finite group and |Γ| its order. Let R(k, Γ) (resp. R m(k, Γ)) 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 Γ. In the present article, we suppose that Γ is realizable as Galois group over k of a Galois extension (resp. tame Galois extension) - e.g. Γ solvable - and the center Z(Γ) of Γ is non-trivial - e.g. Γ nilpotent non-trivial. For each prime divisor p of the order of Z(Γ), we define a natural number n p. We show that if the class number of k is prime to n p, then R(k, Γ) (resp. R m(k, Γ)) is the full group Cl(k). For instance, this result applies to a nilpotent group Γ having even order, with n 2 = |Γ|/2.