Address:
Mohamed Mahmoud Chems-Eddin, Mohammed First University, Mathematics Department, Sciences Faculty, Oujda, Morocco
Abdelmalek Azizi, Mohammed First University, Mathematics Department, Sciences Faculty, Oujda, Morocco
Abdelkader Zekhnini, Mohammed First University, Mathematics Department, Pluridisciplinary faculty, Nador, Morocco
E-mail:
2m.chemseddin@gmail.com
abdelmalekazizi@yahoo.fr
zekha1@yahoo.fr
Abstract: 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 $5\hspace{4.44443pt}(\@mod \; 8)$.
AMSclassification: primary 11R29; secondary 11R11, 11R23, 11R32.
Keywords: cyclotomic \mathbb{Z}_2-extension, 2-rank, 2-class group.
DOI: 10.5817/AM2021-1-13