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 $K=\mathbb{Q}({\zeta }_3)$, 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 $\mathrm{Gal}\left(N^{\ast }/N\right)\cong \mathbb{Z}/3\mathbb{Z}\times \mathbb{Z}/3\mathbb{Z}$ is elementary bicyclic.

22 pages, 2 figures, 8 tables