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Classes de Steinitz dʼextensions galoisiennes à groupe de Galois de centre non trivial

Abstract : Let k be a number field and Cl(k) its class group. Let Γ be a finite group and |Γ| its order. Let R(k, Γ) (resp. R m(k, Γ)) be the subset of Cl(k) consisting of those classes which are realizable as Steinitz classes of Galois extensions (resp. tamely ramified Galois extensions) of k with Galois group isomorphic to Γ. In the present article, we suppose that Γ is realizable as Galois group over k of a Galois extension (resp. tame Galois extension) - e.g. Γ solvable - and the center Z(Γ) of Γ is non-trivial - e.g. Γ nilpotent non-trivial. For each prime divisor p of the order of Z(Γ), we define a natural number n p. We show that if the class number of k is prime to n p, then R(k, Γ) (resp. R m(k, Γ)) is the full group Cl(k). For instance, this result applies to a nilpotent group Γ having even order, with n 2 = |Γ|/2.
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Submitted on : Thursday, April 1, 2021 - 10:57:38 AM
Last modification on : Thursday, September 8, 2022 - 4:13:06 PM

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Bouchaïb Sodaïgui. Classes de Steinitz dʼextensions galoisiennes à groupe de Galois de centre non trivial. Journal of Number Theory, Elsevier, 2013, 133 (2), pp.611-619. ⟨10.1016/j.jnt.2012.08.015⟩. ⟨hal-03187571⟩

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