On the rank of the $2$-class group of $\mathbb{Q}(\sqrt{p},\sqrt{q},\sqrt{-1})$

Abstract

Let $d$ be a square-free integer, $\mathbf{k}=\mathbb{Q}(\sqrt{d},i)$ and $i=\sqrt{-1}$. Let ${\mathbf{k}}_1^{(2)}$ be the Hilbert $2$-class field of $\mathbf{k}$, ${\mathbf{k}}_2^{(2)}$ be the Hilbert $2$-class field of ${\mathbf{k}}_1^{(2)}$ and $G=\mathrm{Gal}({\mathbf{k}}_2^{(2)}/\mathbf{k})$ be the Galois group of ${\mathbf{k}}_2^{(2)}/\mathbf{k}$. Our goal is to give necessary and sufficient conditions to have $G$ metacyclic in the case where $d=pq$, with $p$ and $q$ are primes such that $p\equiv 1(\mathrm{mod}8)$ and $q\equiv 5(\mathrm{mod}8)$ or $p\equiv 1(\mathrm{mod}8)$ and $q\equiv 3(\mathrm{mod}4)$.