Capitulation in the absolutely abelian extensions of some number fields II

TL;DR

This paper investigates the capitulation of 2-ideal classes in certain imaginary biquadratic number fields, providing explicit computations of capitulation kernels and demonstrating that strongly ambiguous classes capitulate in the absolute genus field.

Contribution

It offers explicit calculations of capitulation kernels in a family of imaginary biquadratic fields and shows strong ambiguous classes capitulate in the absolute genus field.

Findings

Capitulation kernels are explicitly computed for three quadratic extensions.

Strongly ambiguous classes in these fields capitulate in the absolute genus field.

The results deepen understanding of ideal class behavior in specific number field extensions.

Abstract

We study the capitulation of $2$-ideal classes of an infinite family of imaginary biquadratic number fields consisting of fields $k =Q(\sqrt{pq_1q_2}, i)$, where $i=\sqrt{-1}$ and $q_1\equiv q_2\equiv-p\equiv-1 \pmod 4$ are different primes. For each of the three quadratic extensions $K/k$ inside the absolute genus field $k^{(*)}$ 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 $k^{(*)}$.