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

Published online by Cambridge University Press:  20 November 2018

Ross Griffiths  and
Mikaël Lescop
Ross Griffiths
Affiliation:
Department of Mathematics and Statistics, McMaster University, 1280 Main StreetWest, Hamilton, Ontario L8S 4K1, e-mail: ross.griffiths@gmail.com
Mikaël Lescop
Affiliation:
IUT de Brest, Départment GMP, Rue de Kergoat, 29200 Brest, France, e-mail: mikael_lescop@hotmail.com
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Abstract

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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.

Keywords

11R7019F15
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 61 , Issue 5 , 01 October 2009 , pp. 1073 - 1091
Copyright
Copyright © Canadian Mathematical Society 2009

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