On the $2$-class group of some number fields with large degree
- 1 January 2021
- journal article
- research article
- Published by Masaryk University Press in Archivum Mathematicum
- Vol. 57 (1), 13-26
- https://doi.org/10.5817/am2021-1-13
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 $5\pmod 8$.
Keywords
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