Classes réalisables d'extensions métacycliques de degré lm

Abstract

Let k be a number field and O k 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 O k -order in the semi-simple algebra k [ Γ ] containing O k [ 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 ⊗ O k [ Γ ] O N ] = c , where O N 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.