On the Hilbert 2-class fields of some real quadratic number fields and applications

Let \(\mathbb {k}=\mathbb {Q}(\sqrt{p_{1}p_{2}q_{1}q_{2}})\) $\mathbb{k}=\mathbb{Q}(\sqrt{p_1{p}_2{q}_1{q}_2})$ be a real quadratic number field, where \(p_{i}\equiv -q_{i}\equiv 1 \pmod 4\) $p_i\equiv -q_i\equiv 1(mod4)$ , \(i=1, 2\) $i=1,2$ , are different prime integers and \(\textrm{C}_{\mathbb {k}, 2}\) ${\text{C}}_{\mathbb{k},2}$ its 2-class group. Let \(\mathbb {k}_2^{(1)}\) ${\mathbb{k}}_2^{(1)}$ (resp. \(\mathbb {k}_2^{(2)}\) ${\mathbb{k}}_2^{(2)}$ ) be the first (resp. second) Hilbert 2-class field of \(\mathbb {k}\) $\mathbb{k}$ . In this article, we investigate the metacyclicity of \(G=Gal(\mathbb {k}_{2}^{(2)}/\mathbb {k})\) $G=Gal({\mathbb{k}}_2^{(2)}/\mathbb{k})$ and the cyclicity of \(G'=Gal(\mathbb {k}_{2}^{(2)}/\mathbb {k}_{2}^{(1)})\) $G^{\prime }=Gal({\mathbb{k}}_2^{(2)}/{\mathbb{k}}_2^{(1)})$ , the derived subgroup of G, assuming \(\textrm{C}_{\mathbb {k}, 2}\simeq G/G'\simeq \mathbb {Z}/2\mathbb {Z}\times \mathbb {Z}/2^{n}\mathbb {Z}\) ${\text{C}}_{\mathbb{k},2}\simeq G/G^{\prime }\simeq \mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2^n\mathbb{Z}$ , with \(n\ge 2\) $n\ge 2$ . As application we study the capitulation of \(\textrm{C}_{\mathbb {k}, 2}\) ${\text{C}}_{\mathbb{k},2}$ in the quadratic and biquadratic subfields of \(\mathbb {k}_{2}^{(1)}\) ${\mathbb{k}}_2^{(1)}$ .