On the $2$-class group of some number fields with large degree

Abstract

Let $d$ be an odd square-free integer, $m\geq 3$, $k$:$=\mathbb{Q}(\sqrt{d}, \sqrt{-1})$, $\mathbb{Q}(\sqrt{-2}, \sqrt{d})$ or $\mathbb{Q}(\sqrt{-2}, \sqrt{-d})$, and $L_{m,d}:=\mathbb{Q}(\zeta_{2^m},\sqrt{d})$. In this paper, we shall determine all the fields $L_{m, d}:=\mathbb{Q}(\zeta_{2^m},\sqrt{d})$, with $m\geq 3$ is an integer, such that the class number of $L_{m, d}$ is odd. Furthermore, using the cyclotomic $\mathbb{Z}_2$-extensions of $k$, we compute the rank of the $2$-class group of $L_{m, d}$ whenever the divisors of $d$ are congruent $3$ or $5\pmod 8$.