Realizable Galois module classes over the group ring for non abelian extensions
Annales de l'Institut Fourier, Volume 63 (2013) no. 1, p. 303-371

Given an algebraic number field k and a finite group Γ, we write (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 (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=VC, 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 () of (O k [Γ]) under extension of scalars from O k [Γ] to a maximal order O k [Γ] in k[Γ]. The main result here thus generalizes the calculation of (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.

Étant donné un corps de nombres k et un groupe fini Γ, on note (O k [Γ]) le sous-ensemble du groupe de classes localement libre Cl(O k [Γ]) formé par les classes d’anneaux d’entiers O N d’extensions galoisiennes modérées N/k avec Gal(N/k)Γ. Nous déterminons (O k [Γ]), et montrons que c’est un sous-groupe de Cl(O k [Γ]), au moyen d’une description utilisant un idéal de Stickelberger et des propriétés de certains codes cycliques, lorsque k contient une racine de l’unité d’ordre premier p et Γ=VC, où V est un groupe élémentaire abélien d’ordre p r et C est un groupe cyclique d’ordre m>1 agissant fidèlement sur V et rendant V un 𝔽 p [C]-module irréductible. Ceci généralise et raffine des résultats de Byott, Greither et Sodaïgui pour p=2 dans Crelle, respectivement de Bruche et Sodaïgui pour p>2 dans J. Number Theory, lesquels couvrent seulement le cas m=p r -1 et déterminent seulement l’image () de (O k [Γ]) sous l’extension des scalaires de O k [Γ] à un ordre maximal O k [Γ] dans k[Γ]. Le résultat principal ici généralise donc le calcul de (O k [A 4 ]) pour le groupe alterné A 4 de degré 4 (le cas p=r=2) donné par Byott et Sodaïgui dans Compositio.

DOI : https://doi.org/10.5802/aif.2762
Classification:  11R33
Keywords: Galois module structure; Rings of algebraic integers; Locally free classgroup; Fröhlich-Lagrange resolvent; Realizable classes; Embedding problem; Stickelberger ideal; Cyclic codes.
@article{AIF_2013__63_1_303_0,
     author = {Byott, Nigel P. and Soda\"\i gui, Boucha\"\i b},
     title = {Realizable Galois module classes over the group ring for non abelian extensions},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {63},
     number = {1},
     year = {2013},
     pages = {303-371},
     doi = {10.5802/aif.2762},
     mrnumber = {3097949},
     zbl = {06177083},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2013__63_1_303_0}
}
Byott, Nigel P.; Sodaïgui, Bouchaïb. Realizable Galois module classes over the group ring for non abelian extensions. Annales de l'Institut Fourier, Volume 63 (2013) no. 1, pp. 303-371. doi : 10.5802/aif.2762. http://www.numdam.org/item/AIF_2013__63_1_303_0/

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