Geodesic
On the $2$-class group of some number fields with large degree
Archivum mathematicum, Tome 57 (2021) no. 1, pp. 13-26 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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 $5\pmod 8$.
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 $5\pmod 8$.
DOI : 10.5817/AM2021-1-13
Classification : 11R11, 11R23, 11R29, 11R32
Keywords: cyclotomic $\mathbb{Z}_2$-extension; $2$-rank; $2$-class group 
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Chems-Eddin, Mohamed Mahmoud; Azizi, Abdelmalek; Zekhnini, Abdelkader. On the $2$-class group of some number fields with large degree. Archivum mathematicum, Tome 57 (2021) no. 1, pp. 13-26. doi: 10.5817/AM2021-1-13

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