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

Abstract

Given an algebraic number field k and a finite group Γ, we write ℛ(O k [Γ]) for the subset of the locally free classgroup Cl(O k [Γ]) consisting of the classes of rings of integers O N in tame Galois extensions N/k with Gal(N/k)≅Γ. We determine ℛ(O k [Γ]), and show it is a subgroup of Cl(O k [Γ]) by means of a description using a Stickelberger ideal and properties of some cyclic codes, when k contains a root of unity of prime order p and Γ=V⋊C, where V is an elementary abelian group of order p r and C is a cyclic group of order m>1 acting faithfully on V and making V into an irreducible 𝔽 p [C]-module. This extends and refines results of Byott, Greither and Sodaïgui for p=2 in Crelle, respectively of Bruche and Sodaïgui for p>2 in J. Number Theory, which cover only the case m=p r -1 and determine only the image ℛ(ℳ) of ℛ(O k [Γ]) under extension of scalars from O k [Γ] to a maximal order ℳ⊃O k [Γ] in k[Γ]. The main result here thus generalizes the calculation of ℛ(O k [A 4 ]) for the alternating group A 4 of degree 4 (the case p=r=2) given by Byott and Sodaïgui in Compositio.