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

Published:2009-10-01
Printed: Oct 2009
• Ross Griffiths
• Mikaël Lescop
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## Abstract

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 $\Q(\sqrt{2},\sqrt{\delta})$ where $\delta$ is a squarefree integer.
 MSC Classifications: 11R70 - $K$-theory of global fields [See also 19Fxx] 19F15 - Symbols and arithmetic [See also 11R37]