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arXiv · 2302.09428

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

A Azizi·M Rezzougui·A Zekhnini
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Abstract

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 $1$.

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A Azizi, M Rezzougui, A Zekhnini. 2023-02-18. Cyclicity of the 2-class group of the first Hilbert 2-class field of some number fields. https://doi.org/10.46298/cm.10983

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