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.
Categories: math.NT