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Structure galoisienne d'anneaux entiers

Abstract : Let k be a number field, O its ring of integers and Γ the alternating group A₄ . Assume that k and Q(j) are linearly disjoint over Q. Let M be a maximal O-order in k[ Γ ] containing O[ Γ ] and C1(M) its classgroup. We denote by R(M) the set of realizable classes, that is, the set of classes c ∈ C1(M) such that there exists a Galois extension N/k with Galois group isomorphic to Γ and the class of M⊗₀Γ equal to c, where O is the ring of integers of N. In this thesis, we determine effectively the elements of R(M) and we prove that R(M) is a subgroup of C1(M). When we try to study R(M), we are confronted with an embedding problem connected with the Steinitz classes, another part of this thesis is the study of Steinitz classes of tetrahedrals extensions and we have study too the case when is the symetric group S₄.
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Submitted on : Thursday, April 8, 2021 - 1:51:39 PM
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Marjory Godin. Structure galoisienne d'anneaux entiers. Théorie des nombres [math.NT]. Université de Valenciennes et du Hainaut-Cambrésis, 2002. Français. ⟨NNT : 2002VALE0016⟩. ⟨tel-03192828⟩



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