Classes réalisables d'extensions non abéliennes

Abstract

Let k be a number field and O_k its ring of integers. Let Γ be a finite group, N/k a Galois extension with Galois group isomorphic to Γ, and O_N the ring of integers of N. 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 classgroup. When N/k is tame (i.e., at most tamely ramified), extension of scalars allows us to assign to O_N the class of , denoted , in Cl(ℳ). 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 . In the present article, we prove, by means of a fairly explicit description, that ℛ(ℳ) is a subgroup of Cl(ℳ) when , where V is an 𝔽₂-vector space of dimension r ≧ 2, C a cyclic group of order 2^r − 1, and p a faithful representation of C in V; an example is the alternating group A₄. In the proof, we use some properties of the binary Hamming code and solve an embedding problem connected with Steinitz classes. In addition, we determine the set of Steinitz classes of tame Galois extensions of k, with the above group as Galois group, and prove that it is a subgroup of the classgroup of k.