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 G be some metabelian 2-group satisfying the condition G/G’ ≃ ℤ/2ℤ × ℤ/2ℤ × ℤ/2ℤ. In this paper, we construct all the subgroups of G of index 2 or 4, we give the abelianization types of these subgroups and we compute the kernel of the transfer map. Then we apply these results to study the capitulation problem for the 2-ideal classes of some fields k satisfying the condition $Gal(k{₂}^{\left(2\right)}/k)\simeq G$, where $k{₂}^{\left(2\right)}$ is the second Hilbert 2-class field of k.

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 $5(mod8)$.

For any square-free positive integer $m\equiv 10(mod16)$ with $m\ge 26$, we prove that the class number of the real cyclotomic field $\mathbb{Q}({\zeta }_{4m}+{\zeta }_{4m}^{-1})$ is greater than $1$, where ${\zeta }_{4m}$ is a primitive $4m$th root of unity.

We present algorithms for the computation of extreme binary Humbert forms in real quadratic number fields. With these algorithms we are able to compute extreme Humbert forms for the number fields $\mathbb{Q}\left(\sqrt{13}\right)$ and $\mathbb{Q}\left(\sqrt{17}\right)$. Finally we compute the Hermite-Humbert constant for the number field $\mathbb{Q}\left(\sqrt{13}\right)$.

We describe an algorithm due to Gauss, Shanks and Lagarias that, given a non-square integer $D\equiv 0,1$ mod $4$ and the factorization of $D$, computes the structure of the $2$-Sylow subgroup of the class group of the quadratic order of discriminant $D$ in random polynomial time in $log\left|D\right|$.