Classes réalisables d'extensions non abéliennes de degré [formula omitted

Abstract

Let k be a number field and Ok its ring of integers. Let p be an odd prime number. Let Γ be a non-abelian group of order p3. Let M be a maximal Ok-order in the semi-simple algebra k[Γ] containing Ok[Γ], and let Cl(M) be its locally free classgroup. We define the set R(M) of realizable classes to be the set of classes c∈Cl(M) such that there exists a Galois extension N/k which is tame, with Galois group isomorphic to Γ, and for which the class of M⊗Ok[Γ]ON is equal to c, where ON is the ring of integers of N. Let ξ (resp. ξp2) be a primitive pth (resp. p2th) root of unity. In the present article, under the hypothesis that k/Q and Q(ξ)/Q are linearly disjoint and k(ξp2)/k(ξ) is not ramified when Γ has exponent p2, we define a subset of R(M) by means of a Stickelberger ideal, and prove that it is a subgroup of Cl(M) contained in R(M).