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Classes de Steinitz et classes galoisiennes réalisables d'extensions non abéliennes

Abstract : Let k be a number field, Cl(k) its class group and Ok its ring of integers. Let Rm(k;Γ) be the subset of Cl(k) consisting of those classes which are realizable as Steinitz classes of tame Galois extensions of k with Galois group isomorphic to Γ. LetMbe a maximal Ok-order in the semi-simple algebra k[Γ ] containing Ok[Γ], and Cl(M) its locally free classgroup. We define the set R(M) of realizable Galois module classes to be the set of classes c 2 Cl(M) such that there exists a Galois extension N=k which is tame, with Galois group isomorphic to Γ, and for which [MOk[ô] ON] = c, where ON is the ring of integers of N. When Γ is a nonabelian group of order 16 or an extra-special group of order 32, we show that Rm(k; Γ) is the full group Cl(k) if the class number of k is odd, with the hypothesis i 2 k for the modular group of order 16. When Γ = C oH, where C (resp. H) is a cyclic group of order l (resp. m), l is prime and H acting faithfully on C, we define a subset of R(M) and prove, by means of a description using a Stickelberger ideal, that it is a subgroup of Cl(M), under the hypothesis that k and the l-th cyclotomic field over Q are linearly disjoint.
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Farah Sbeity. Classes de Steinitz et classes galoisiennes réalisables d'extensions non abéliennes. Mathématiques [math]. Université de Valenciennes et du Hainaut-Cambrésis, 2010. Français. ⟨NNT : 2010VALE0018⟩. ⟨tel-03004776⟩

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