Capitulation in the absolutely abelian extensions of some number fields II

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 $q_1\equiv q_2\equiv -p\equiv -1(\mathrm{mod}4)$ are different primes. For each of the three quadratic extensions $K/k$ inside the absolute genus field $k^{(\ast )}$ 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 $k^{(\ast )}$.

17 pages, To be published by Acta Mathematica Vietnamica, 2016. arXiv admin note: substantial text overlap with arXiv:1507.00295, arXiv:1503.01992