Abdelkader Zekhnini, Abdelmalek Azizi, Mohammed Taous

Capitulation in the absolutely abelian extensions of some fields Q(\sqrt{p_1p_2q}, i)

We study the capitulation of $2$-ideal classes of an infinite family of imaginary bicyclic biquadratic number fields consisting of fields $\kk =\QQ(\sqrt{p_1p_2q}, i)$, where $i=\sqrt{-1}$ and $p_1\equiv p_2\equiv-q\equiv1 \pmod 4$ are different primes. For each of the three quadratic extensions $\KK/\kk$ inside the absolute genus field $\kk^{(*)}$ of $\kk$, we compute the capitulation kernel of $\KK/\kk$. Then we deduce that each strongly ambiguous class of $\kk/\QQ(i)$ capitulates already in $\kk^{(*)}$, which is smaller than the relative genus field $(\kk/\QQ(i))^*$.

Accéder au lien