Classes réalisables d'extensions métacycliques de degré lm

AbstractLet k be a number field and Ok its ring of integers. Let l be a prime number and m a natural number. Let C (resp. H) be a cyclic group of order l (resp. m). Let Γ=C⋊H be a metacyclic group of order lm, with H acting faithfully on C. Let M be a maximal Ok-order in the semi-simple algebra k[Γ] containing Ok[Γ], and Cl(M) its locally free class group. 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 [M⊗Ok[Γ]ON]=c, where ON is the ring of integers of N. In the present article, we define a subset of R(M) and prove, by means of a description using a Stickelberger ideal, that it is a subgroup of Cl(M), under the hypothesis that k and the l-th cyclotomic field over Q are linearly disjoint