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Capitulation in the Absolutely Abelian Extensions of some Number Fields II

Abdelmalek Azizi , $icon-email$ Abdelkader Zekhnini , Mohammed Taous

Abstract

We study the capitulation of 2-ideal classes of an infinite family of imaginary biquadratic number fields consisting of fields $k = Q (\sqrt{p q_{1} q_{2}}, i)$ , where $i = \sqrt{- 1}$ and $q_{1} \equiv q_{2} \equiv - p \equiv - 1$ (mod 4) are different primes. For each of the three quadratic extensions $K / k$ inside the absolute genus field $k^{(*)}$ of $k$ , we compute the capitulation kernel of $K / k$ . Then we deduce that each strongly ambiguous class of $k / Q (i)$ capitulates already in $k^{(*)}$ .

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Capitulation in the Absolutely Abelian Extensions of some Number Fields II

Abstract