## A Azizi ; M Rezzougui ; A Zekhnini - Cyclicity of the 2-class group of the first Hilbert 2-class field of some number fields

cm:10983 - Communications in Mathematics, March 1, 2023, Volume 32 (2024), Issue 1 - https://doi.org/10.46298/cm.10983
Cyclicity of the 2-class group of the first Hilbert 2-class field of some number fields

Authors: A Azizi ; M Rezzougui ; A Zekhnini

Let $\mathds{k}$ be a real quadratic number field. Denote by $\mathrm{Cl}_2(\mathds{k})$ its $2$-class group and by $\mathds{k}_2^{(1)}$ (resp. $\mathds{k}_2^{(2)}$) its first (resp. second) Hilbert $2$-class field. The aim of this paper is to study, for a real quadratic number field whose discriminant is divisible by one prime number congruent to $3$ modulo 4, the metacyclicity of $G=\mathrm{Gal}(\mathds{k}_2^{(2)}/\mathds{k})$ and the cyclicity of $\mathrm{Gal}(\mathds{k}_2^{(2)}/\mathds{k}_2^{(1)})$ whenever the rank of $\mathrm{Cl}_2(\mathds{k})$ is $2$, and the $4$-rank of $\mathrm{Cl}_2(\mathds{k})$ is $1$.

Volume: Volume 32 (2024), Issue 1
Published on: March 1, 2023
Accepted on: February 21, 2023
Submitted on: February 21, 2023
Keywords: Mathematics - Number Theory