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Article Dans Une Revue Annales de l'Institut Fourier Année : 2013

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

Classes galoisiennes réalisables sur l’anneau de groupe d’extensions non abéliennes

Résumé

Given an algebraic number field k and a finite group T, we write R(Ok[T]) for the subset of the locally free classgroup Cl(Ok[T]) consisting of the classes of rings of integers ON in tame Galois extensions N/k with Gal(N/k) T = T. We determine R(Ok[T]), and show it is a subgroup of Cl(Ok[T]) 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 T = V oC, where V is an elementary abelian group of order pr and C is a cyclic group of order m > 1 acting faithfully on V and making V into an irreducible Fp[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 = pr -1 and determine only the image R(M) of R(Ok[T]) under extension of scalars from Ok[T] to a maximal orderM T Ok[U] in k[T]. The main result here thus generalizes the calculation of R(Ok[A4]) for the alternating group A4 of degree 4 (the case p = r = 2) given by Byott and Sodaïgui in Compositio.

Dates et versions

hal-03165412 , version 1 (10-03-2021)

Identifiants

Citer

Nigel Byott, Bouchaïb Sodaïgui. Realizable Galois module classes over the group ring for non abelian extensions. Annales de l'Institut Fourier, 2013, 63 (1), p. 303-371. ⟨10.5802/aif.2762⟩. ⟨hal-03165412⟩
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