Structure of relative genus fields of cubic Kummer extensions

TL;DR

This paper investigates the structure of relative genus fields in cubic Kummer extensions over cyclotomic fields, identifying conditions under which the genus group is elementary bicyclic.

Contribution

It characterizes all radicands and conductors for which the genus group in these extensions is elementary bicyclic.

Findings

Identifies all D and f with a bicyclic genus group

Provides explicit conditions for the genus group's structure

Enhances understanding of genus fields in cubic Kummer extensions

Abstract

Let $N=K(\sqrt[3]{D})$ be a cubic Kummer extension of the cyclotomic field $K=\mathbb{Q}(\zeta_3)$, containing a primitive cube root of unity $\zeta_3$, with cube free integer radicand $D>1$. Denote by $f$ the conductor of the abelian extension $N/K$, and by $N^{\ast}$ the relative genus field of $N/K$. The aim of the present work is to find out all positive integers $D$ and conductors $f$ such that the genus group $\operatorname{Gal}\left(N^{\ast}/N\right)\cong \mathbb{Z}/3\mathbb{Z}\times\mathbb{Z}/3\mathbb{Z}$ is elementary bicyclic.