Capitulation in the absolutely abelian extensions of some fields $\mathbb{Q}(\sqrt{p_1{p}_{2}q},\sqrt{-1})$

Abstract

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

18 pages. arXiv admin note: substantial text overlap with arXiv:1503.01992