Beschreibung:
<jats:p><jats:italic>Abstract.</jats:italic> Let <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0008414X00003850_inline1" /><jats:tex-math>$E/F$</jats:tex-math></jats:alternatives></jats:inline-formula> 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 <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0008414X00003850_inline2" /><jats:tex-math>$E$</jats:tex-math></jats:alternatives></jats:inline-formula> provided that the 2-primary Hilbert kernel of <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0008414X00003850_inline3" /><jats:tex-math>$F$</jats:tex-math></jats:alternatives></jats:inline-formula> 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 <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0008414X00003850_inline2" /><jats:tex-math>$E$</jats:tex-math></jats:alternatives></jats:inline-formula> whenever <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0008414X00003850_inline3" /><jats:tex-math>$F$</jats:tex-math></jats:alternatives></jats:inline-formula> 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 <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0008414X00003850_inline4" /><jats:tex-math>$\mathbb{Q}(\sqrt{2},\sqrt{\delta )}$</jats:tex-math></jats:alternatives></jats:inline-formula> where <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0008414X00003850_inline5" /><jats:tex-math>$\delta $</jats:tex-math></jats:alternatives></jats:inline-formula> is a squarefree integer.</jats:p>