Cyclicity of the 2-class group of the first Hilbert 2-class field of
some number fields

Authors: A Azizi ; M Rezzougui ; A Zekhnini

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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 $1$.