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 $p_1\equiv p_2\equiv -q\equiv1 \pmod4$ be different primes such that $\displaystyle\left(\frac{2}{p_1}\right)= \displaystyle\left(\frac{2}{p_2}\right)=\displaystyle\left(\frac{p_1}{q}\right)=\displaystyle\left(\frac{p_2}{q}\right)=-1$. Put $d=p_1p_2q$ 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.