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Les corps multi-quadratiques p-rationnels

Abstract : For each prime p, we prove the existence of infinitely many real quadratic p-rational number fields as well as the existence of a real and an imaginary bi-quadratic p-rational number field. Moreover for p = 3, we show the existence of infinitely many imaginary bi-quadratic 3-rational number fields. For p > 5 and F a real multi-quadratic p-rational number field with tame kernel of ordre prime to p, we prove the existence of infinitely many imaginary quadratic extensions of F, p-rational.Using a recent method developed by Greenberg, we deduce the existence of Galois extensions of Q whose Galois groups are isomorphic to open subgroups of GLn(Zp) for n = 4 and n = 5 and at least for all p ≤ 718.328.637. Finally, we give a new reformulation of the conjectures of Ankeny-Artin-Chowla and Mordell, in terms of the p-rationality of Q(√p).
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Submitted on : Monday, February 7, 2022 - 10:54:08 AM
Last modification on : Thursday, September 8, 2022 - 4:00:43 AM
Long-term archiving on: : Sunday, May 8, 2022 - 6:26:00 PM


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  • HAL Id : tel-03559781, version 1



Youssef Benmerieme. Les corps multi-quadratiques p-rationnels. Algèbre commutative [math.AC]. Université de Limoges, 2021. Français. ⟨NNT : 2021LIMO0100⟩. ⟨tel-03559781⟩



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