Realisable classes, Stickelberger subgroup and its behaviour under change of the base field
Publications Mathématiques de Besançon (2015), pp. 69-92.

Soient K un corps de nombres d’anneau des entiers O K et G un groupe fini. On note R(O K [G]) l’ensemble des classes dans le groupe des classes des modules localement libres Cl(O K [G]) qui peuvent être obtenues par l’anneau des entiers des K-algèbres galoisiennes modérément ramifiées de groupe de Galois G. McCulloh a prouvé que, pour tout G, l’ensemble R(O K [G]) est contenu dans le soi-disant sous-groupe de Stickelberger St(O K [G]) dans Cl(O K [G]).

Dans ce papier d’abord nous nous focalisons sur la relation entre St(O K [G]) et Cl (O K [G]), où Cl (O K [G]) est le noyau du morphisme Cl(O K [G])Cl(O K ), induit par l’augmentation O K [G]O K . Puis, comme exemple de calcul du groupe St(O K [G]), nous montrons, en utilisant sa définition, que St([G]) est trivial si G est soit un groupe cyclique d’ordre p soit un groupe diédral d’ordre 2p, avec p premier impair.

Enfin, nous montrons la fonctorialité de St(O K [G]) par rapport au changement du corps de base. Ceci implique que, soit L est un corps de nombres, si N est une L-algèbre galoisienne modérément ramifiée, de groupe de Galois G, et St(O K [G]) est connu être trivial pour un certain sous-corps K de L, alors O N est un O K [G]-module stablement libre.

Let K be an algebraic number field with ring of integers O K and let G be a finite group. We denote by R(O K [G]) the set of classes in the locally free class group Cl(O K [G]) realisable by rings of integers in tamely ramified G-Galois K-algebras. McCulloh showed that, for every G, the set R(O K [G]) is contained in the so-called Stickelberger subgroup St(O K [G]) of Cl(O K [G]).

In this paper first we describe the relation between St(O K [G]) and Cl (O K [G]), where Cl (O K [G]) is the kernel of the morphism Cl(O K [G])Cl(O K ), induced by the augmentation map O K [G]O K . Then, as an example of computation of St(O K [G]), we show, just using its definition, that St([G]) is trivial, when G is a cyclic group of order p or a dihedral group of order 2p, where p is an odd prime number.

Finally we prove that St(O K [G]) has good functorial behaviour under change of the base field. This has the interesting consequence that, given an algebraic number field L, if N is a tame Galois L-algebra with Galois group G and St(O K [G]) is known to be trivial for some subfield K of L, then O N is stably free as an O K [G]-module.

Reçu le : 2014-07-14
Publié le : 2016-01-24
DOI : https://doi.org/10.5802/pmb.13
Classification : 11R33,  11R04,  11R18,  11R29,  11R32,  11R65
Mots clés: Galois module structure, Realisable classes, Locally free class groups, Fröhlich’s Hom-description of locally free class groups, Stickelberger’s theorem
@article{PMB_2015____69_0,
     author = {Andrea Siviero},
     title = {Realisable classes, Stickelberger subgroup and its behaviour under change of the base field},
     journal = {Publications Math\'ematiques de Besan\c con},
     publisher = {Presses universitaires de Franche-Comt\'e},
     year = {2015},
     pages = {69-92},
     doi = {10.5802/pmb.13},
     zbl = {1414.11152},
     language = {en},
     url = {pmb.centre-mersenne.org/item/PMB_2015____69_0/}
}
Andrea Siviero. Realisable classes, Stickelberger subgroup and its behaviour under change of the base field. Publications Mathématiques de Besançon (2015), pp. 69-92. doi : 10.5802/pmb.13. https://pmb.centre-mersenne.org/item/PMB_2015____69_0/

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