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

Byott, Nigel P.; Sodaı̈gui, Bouchaı̈b

doi:10.5802/aif.2762

Cited by 6 publications

(3 citation statements)

References 12 publications

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“…Indeed, when G is non-abelian and K = Q, determining if R(O K , the dihedral group of order 8, with the assumption that the ray class group modulo 4O K of O K has odd order (see [BS05a]); and G = A 4 , without any restriction on the base field K (see [BS05b]). Recently in [BS13], Nigel P. Byott and Bouchaïb Sodaïgui, under the assumption that K contains a root of unity of prime order p, showed that R(O K [G]) is a subgroup of Cl(O K [G]), when G is the semidirect product V C of an elementary abelian group V of order p r by any non-trivial cyclic group C which acts faithfully on V and makes V into an irreducible F p [C]-module (where F p is the finite field with p elements). This last result contains as a corollary the main result of [BS05b].…”

Section: Theorem 12 -For Every Algebraic Number Field K and Finite mentioning

confidence: 99%

Realisable classes, Stickelberger subgroup and its behaviour under change of the base field

Siviero¹

2016

Publications Mathématiques De Besançon. Algèbre Et Théorie Des Nombres

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Section: Theorem 12 -For Every Algebraic Number Field K and Finite mentioning

confidence: 99%

Realisable classes, Stickelberger subgroup and its behaviour under change of the base field

Siviero¹

2016

Publications Mathématiques De Besançon. Algèbre Et Théorie Des Nombres

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“…We point out that when C is abelian, RðO, O k ½CÞ is a subgroup of ClðO k ½CÞ according to [10]. (If C is not abelian, one may see for instance [2,4] and their references for works on RðO, O k ½CÞ and RðO, MÞ:)…”

Section: Introductionmentioning

confidence: 99%

On realizable Galois module classes by the inverse different

Sodaı̈gui

Taous

2020

Communications in Algebra

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Let k be a number field and O k its ring of integers. Let C be a finite group. Let M be a maximal O k-order in the semi-simple algebra k½C containing O k ½C: Let ClðO k ½CÞ (resp. ClðMÞ) be the locally free classgroup of O k ½C (resp. M). We denote by RðD À1 , O k ½CÞ (resp. RðD À1 , MÞ) the set of classes c in ClðO k ½CÞ (resp. ClðMÞ) such that there exists a tamely ramified extension N/k, with Galois group isomorphic to C (C-extension) and the class of D À1 N=k (resp. M O k ½C D À1 N=k) is equal to c, where D N=k is the different of N/k and D À1 N=k its inverse different. We say that RðD À1 , O k ½CÞ (resp. RðD À1 , MÞ) is the set of realizable Galois module classes by the inverse different. In the present article, combining some of our published results, and a result due to A. Fr€ ohlich giving a link between the Galois module class of the ring of integers of a tamely ramified C-extension and that of its inverse different, we explicitly determine RðD À1 , O k ½CÞ (resp. RðD À1 , MÞ) for several groups C and show that it is a subgroup of ClðO k ½CÞ (resp. ClðMÞ). In addition, we determine the set of the Steinitz classes of D À1 N=k , N/k runs through the set of tamely ramified C-extension of k, and prove that is a subgroup of Cl(k), also for several groups C: ARTICLE HISTORY

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“…Pour des résultats récents dans la direction de l'étude de la conjecture non abélienne sur les classes réalisables voir [3][4][5][6][7]14,19].…”

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