On the strongly ambiguous classes of some biquadratic number fields

Abstract

We study the capitulation of ideal classes in an infinite family of imaginary bicyclic biquadratic number fields consisting of fields $k=Q(\sqrt{2pq},i)$, where $i=\sqrt{-1}$ and $p\equiv -q\equiv 1(\mathrm{mod}4)$ 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 $k^{(\ast )}$, which is smaller than the relative genus field ${\left(k/Q(i)\right)}^{\ast }$.