On the rank of the $2$-class group of some imaginary triquadratic number fields

Let $d$ be an odd square-free integer and ${\zeta }_8$ a primitive $8$-th root of unity. The purpose of this paper is to investigate the rank of the $2$-class group of the fields $L_d=\mathbb{Q}({\zeta }_8,\sqrt{d})$.