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. 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.

Realizable Galois module classes over the group ring for non abelian extensions

“…An analogous result to that of [4] for the corresponding metabelian groups of order p r (p r − 1) (where p is an odd TOME 63 (2013), FASCICULE 1 prime) was given in [1] under the hypothesis ξ p ∈ k. In both [1] and [4] we drew on the language of coding theory: the construction of Γ-extensions can be conveniently described in terms of a certain cyclic code of length p r − 1 (for p = 2 and p > 2 respectively) over the field F p . Our goal in this paper is to improve on the results in [1] and [4] in two directions, treating the cases p = 2 and p > 2 simultaneously. We again assume that ξ p ∈ k. The first improvement is that we will work over the group ring O k [Γ] rather than over a maximal order M, so that we obtain a description of R(O k [Γ]) itself and not just of its image R(M).…”

“…In the excluded case m = 1, the assertions of Theorems 2 and 3 and the first equality of Theorem 1 are trivially true and the second equality of Theorem 1 reduces to McCulloh's result (1).…”

“…We reformulate this result in a more concrete form, allowing easier comparison with [1] and [4], at the end of this paper (see Theorem 7.3.1).…”

“…The proof of the above results will occupy most of the paper ( § §2-6), and draws quite heavily on the methods of [1] and [4]. In particular, we shall again use the language of cyclic codes.…”

“…A fruitful approach is to extend scalars from O k [Γ] to a maximal order M ⊃ O k [Γ] in k [Γ], and then to investigate the image R(M) of R(O k [Γ]) in Cl(M). This has been done in a number of cases [1,4,15,22,23,24,25,26]. In particular, nonabelian groups Γ of order lq, with l > q both prime, were considered in [23] under the assumption that k ∩ Q(ξ lq ) = Q.…”