On the 2-Rank of the Hilbert Kernel of Number Fields

Griffiths, Ross ; Lescop, Mikaël

Canadian journal of mathematics, Tome 61 (2009) no. 5, pp. 1073-1091

Résumé

Let $E/F$ be a quadratic extension of number fields. In this paper, we show that the genus formula for Hilbert kernels, proved by M. Kolster and A. Movahhedi, gives the 2-rank of the Hilbert kernel of $E$ provided that the 2-primary Hilbert kernel of $F$ is trivial. However, since the original genus formula is not explicit enough in a very particular case, we first develop a refinement of this formula in order to employ it in the calculation of the 2-rank of $E$ whenever $F$ is totally real with trivial 2-primary Hilbert kernel. Finally, we apply our results to quadratic, bi-quadratic, and tri-quadratic fields which include a complete 2-rank formula for the family of fields $\mathbb{Q}(\sqrt{2},\sqrt{\delta )}$ where $\delta $ is a squarefree integer.

Détail

DOI : 10.4153/CJM-2009-051-3

Mots-clés : 11R70, 19F15

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