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

TL;DR

This paper investigates the structure and cyclicity properties of the 2-class groups and Galois groups associated with the first and second Hilbert 2-class fields of certain real quadratic number fields, focusing on cases with specific discriminant divisibility and class group rank.

Contribution

It provides new results on the metacyclicity of the Galois group and the cyclicity of certain subgroups for real quadratic fields with particular discriminant and class group properties.

Findings

Galois group $ ext{Gal}( ext{k}_2^{(2)}/ ext{k})$ is metacyclic under specified conditions.

The Galois group $ ext{Gal}( ext{k}_2^{(2)}/ ext{k}_2^{(1)})$ is cyclic in the studied cases.

Results apply to fields with discriminant divisible by a prime $ eq 1 mod 4$ and class group rank 2, 4-rank 1.

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 $1$.