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Capitulation in the Absolutely Abelian Extensions of some Number Fields II
Abdelmalek Azizi
,
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 $\Bbb k =\mathbb{Q}(\sqrt {pq_{1}q_{2}}, i)$, 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 $\mathbb K/\Bbb k$ inside the absolute genus field $\Bbb k^{(*)}$ of $\Bbb k$, we compute the capitulation kernel of $\mathbb K/\Bbb k$. Then we deduce that each strongly ambiguous class of $\Bbb k/\mathbb Q(i)$ capitulates already in $\Bbb k^{(*)}$.