On realizable Galois module classes and Steinitz classes of nonabelian extensions

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 semisimple algebra k [ Γ ] containing O k [ Γ ] , and Cl ( M ) its locally free class group. When N / k is tame (i.e., at most tamely ramified), extension of scalars allows us to assign to O N the class of M ⊗ O k [ Γ ] O N , denoted [ M ⊗ O k [ Γ ] O N ] , 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 [ M ⊗ O k [ Γ ] O N ] = c . Let p be an odd prime number and let ξ p be a primitive pth root of unity. In the present article, we prove, by means of a fairly explicit description, that R ( M ) is a subgroup of Cl ( M ) when ξ p ∈ k and Γ = V ⋊ ρ C , where V is an F p -vector space of dimension r ⩾ 1 , C a cyclic group of order p r − 1 , and ρ a faithful representation of C in V; an example is the symmetric group S 3 . In the proof, we use some properties of a cyclic 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 class group of k.