On some imaginary triquadratic number fields $k$ with ${\mathrm{Cl}}_2(k)\simeq (2,4)$ or $(2,2,2)$

Abstract

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, ${\mathrm{Cl}}_2(L_d)$, of $L_d$ is of type $(2,4)$ or $(2,2,2)$.

14 pages