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Let $d$ be a square free integer and $L_d:=\mathbb{Q}({\zeta }_8,\sqrt{d})$. In the present work we determine all the fields $L_d$ such that the $2$-class group, ${\mathrm{Cl}}_2\left(L_d\right)$, of $L_d$ is of type $(2,4)$ or $(2,2,2)$.

Let $d$ be an odd square-free integer, $m\ge 3$ any integer and $L_{m,d}:=\mathbb{Q}({\zeta }_{2^m},\sqrt{d})$. In this paper, we shall determine all the fields $L_{m,d}$ having an odd class number. Furthermore, using the cyclotomic ${\mathbb{Z}}_2$-extensions of some number fields, we compute the rank of the $2$-class group of $L_{m,d}$ whenever the prime divisors of $d$ are congruent to $3$ or $5(mod8)$.

Let $𝕜=\mathbb{Q}\left(\sqrt{2},\sqrt{d}\right)$ be an imaginary bicyclic biquadratic number field, where $d$ is an odd negative square-free integer and ${𝕜}_2^{\left(2\right)}$ its second Hilbert $2$-class field. Denote by $G=\mathrm{Gal}({𝕜}_2^{\left(2\right)}/𝕜)$ the Galois group of ${𝕜}_2^{\left(2\right)}/𝕜$. The purpose of this note is to investigate the Hilbert $2$-class field tower of $𝕜$ and then deduce the structure of $G$.

Let $k$ be a number field with a 2-class group isomorphic to the Klein four-group. The aim of this paper is to give a characterization of capitulation types using group properties. Furthermore, as applications, we determine the structure of the second 2-class groups of some special Dirichlet fields $𝕜=\mathbb{Q}(\sqrt{d},\sqrt{-1})$, which leads to a correction of some parts in the main results of A. Azizi and A. Zekhini (2020).