On the strongly ambiguous classes of $k/Q(\sqrt{-1})$ where $k= Q(\sqrt{2p_1p_2},\sqrt{-1})$

TL;DR

This paper constructs an infinite family of imaginary bicyclic biquadratic fields with complex class group behavior, showing how certain ideal classes capitulate in specific genus fields and analyzing their capitulation in quadratic extensions.

Contribution

It introduces a new family of fields with large 2-class groups and details the capitulation behavior of strongly ambiguous classes within their genus fields.

Findings

Existence of infinite family of fields with 2-class group rank ≥ 3

Strongly ambiguous classes capitulate in the absolute genus field

Detailed analysis of capitulation in quadratic extensions

Abstract

We construct an infinite family of imaginary bicyclic biquadratic number fields $k$ with the 2-ranks of their 2-class groups are $\geq3$, whose strongly ambiguous classes of $k/Q(i)$ capitulate in the absolute genus field $k^{(*)}$, which is strictly included in the relative genus field $(k/Q(i))^*$ and we study the capitulation of the $2$-ideal classes of $k$ in its quadratic extensions included in $k^{(*)}$.