The rank of the 2-class group of some fields with large degree

Mohamed Mahmoud Chems-Eddin

Published 2020-01-03Version 1

Let $d$ be an odd square-free integer, $k= \mathbb{Q}(\sqrt{d}, \sqrt{-1})$ and $L_{n,d}=\mathbb{Q}(\zeta_{2^n},\sqrt{d})$, with $n\geq 3$ is an integer. We compute the rank of the $2$-class group of $L_{n,d}$ when all the divisors of $d$ are congruent to $9\pmod{16}$. Furthermore, we give the rank of the $2$-class group of $ L_{n,d}$ according to the one of $ L_{4,d}$, when the divisors of $d$ are congruent to $7$ or $9\pmod{16}$.