For some real quadratic numbers fields \(\mathbb {k}\) , we aim, in this note, to give necessary and sufficient criteria for the 2-class group of \(\mathbb {k}_2^{(1)}\) , the first Hilbert 2-class field of \(\mathbb {k}\) , to be cyclic. For this, we consider \(\mathbb {k}=\mathbb {Q}(\sqrt{2pq_{1}q_{2}})\) , where \(p\equiv -q_{i}\equiv 1 \pmod 4\) , \(i=1, 2\) , are distinct prime integers, denote respectively by \(\textrm{C}_{\mathbb {k}, 2}\) , \(\mathbb {k}_2^{(1)}\) and \(\mathbb {k}_2^{(2)}\) the 2-class group, the first and the second Hilbert 2-class field of \(\mathbb {k}\) . Put \(G=Gal(\mathbb {k}_{2}^{(2)}/\mathbb {k})\) , the Galois group of \(\mathbb {k}_{2}^{(2)}/\mathbb {k}\) , and \(G'=Gal(\mathbb {k}_{2}^{(2)}/\mathbb {k}_{2}^{(1)})\) its derived subgroup. We are interested in studying the 4-rank of \(\textrm{C}_{\mathbb {k}, 2}\) , the metacyclicity of G and the cyclicity of \(G'\) whenever the 4-rank of \(\textrm{C}_{\mathbb {k}, 2}\simeq G/G'\) is 1. Moreover, in the metacyclic case, we investigate the capitulation of \(\textrm{C}_{\mathbb {k}, 2}\) in the unramified quadratic and biquadratic extensions of \(\mathbb {k}\) .
