Structure of relative genus fields of cubic Kummer extensions

Abstract

Let $N=K(\sqrt[3]{D})$ be a cubic Kummer extension of the cyclotomic field $K=\mathbb{Q}(ζ_3)$, containing a primitive cube root of unity $ζ_3$, 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.