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Sur la borne inférieure du rang du 2-groupe de classes de certains corps multiquadratiques

Published online by Cambridge University Press:  20 November 2018

A. Mouhib
A. Mouhib*
Affiliation:
Univ. Mohammed Ben Abdellah, Faculté polydisciplinaire, Laboratoire d’Informatique, Mathématiques, Automatique et Optoélectronique, B/P 1223, Taza-Gare, Maroccourriel: mouhibali@yahoo.fr
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Résumé

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Soient ${{p}_{1}},\,{{p}_{2}},\,{{p}_{3}}$ et $q$ des nombres premiers distincts tels que ${{p}_{1}}\,\equiv \,{{p}_{2}}\,\equiv \,{{p}_{3}}\,\equiv \,-q\,\equiv \,1\,(\bmod \,4)$, $k=\mathbf{Q}(\sqrt{{{p}_{1}}},\sqrt{{{p}_{2}}},\sqrt{{{p}_{3}}},\sqrt{q})$ et $\text{C}{{\text{l}}_{2}}(k)$ le 2-groupe de classes de $k$. A. Fröhlich a démontré que $\text{C}{{\text{l}}_{2}}(k)$ n’est jamais trivial. Dans cet article, nous donnons une extension de ce résultat, en démontrant que le rang de $\text{C}{{\text{l}}_{2}}(k)$ est toujours supérieur ou égal à 2. Nous démontrons aussi, que la valeur 2 est optimale pour une famille infinie de corps $k$.

Abstract

Abstract

Let ${{p}_{1}},\,{{p}_{2}},\,{{p}_{3}}$ and $q$ be distinct prime numbers such that ${{p}_{1}}\,\equiv \,{{p}_{2}}\,\equiv \,{{p}_{3}}\,\equiv \,-q\,\equiv \,1\,(\bmod \,4)$, $k=\mathbf{Q}(\sqrt{{{p}_{1}}},\sqrt{{{p}_{2}}},\sqrt{{{p}_{3}}},\sqrt{q})$ and $\text{C}{{\text{l}}_{2}}(k)$ the 2-class group of $k$. A. Fröhlich has shown that $\text{C}{{\text{l}}_{2}}(k)$ can never be trivial. In this article, we give an extension of this result by proving that the rank of $\text{C}{{\text{l}}_{2}}(k)$ is greater or equal to 2. Moreover, we prove that there exist infinitely many fields $k$ in which the rank of $\text{C}{{\text{l}}_{2}}(k)$ is equal to 2.

Keywords

11R2911R11class groupunitsmultiquadratic number fields
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 54 , Issue 2 , 01 June 2011 , pp. 330 - 337
Copyright
Copyright © Canadian Mathematical Society 2011

References

Références

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