Principalization of $2$-class groups of type $(2,2,2)$ of biquadratic fields $\mathbb{Q}\left(\sqrt{\strut p_1p_2q},\sqrt{\strut -1}\right)$

Abstract

Let $p_1\equiv p_2\equiv -q\equiv 1(\mathrm{mod}4)$ be different primes such that $\left(\frac{2}{p_1}\right)=\left(\frac{2}{p_2}\right)=\left(\frac{p_1}{q}\right)=\left(\frac{p_2}{q}\right)=-1$. Put $d=p_1{p}_{2}q$ and $i=\sqrt{-1}$, then the bicyclic biquadratic field $k=\mathbb{Q}(\sqrt{d},i)$ has an elementary abelian $2$-class group, $\mathbf{C}{l}_2(k)$, of rank $3$. In this paper, we study the principalization of the $2$-classes of $k$ in its fourteen unramified abelian extensions ${\mathbb{K}}_j$ and ${\mathbb{L}}_j$ within $k_2^{(1)}$, that is the Hilbert $2$-class field of $k$. We determine the nilpotency class, the coclass, generators and the structure of the metabelian Galois group $G=\mathrm{Gal}(\mathbb{L}/k)$ of the second Hilbert 2-class field $k_2^{(2)}$ of $k$. Additionally, the abelian type invariants of the groups $\mathbf{C}{l}_2({\mathbb{K}}_j)$ and $\mathbf{C}{l}_2({\mathbb{L}}_j)$ and the length of the $2$-class tower of $k$ are given.