Fields $\mathbb{Q}(\sqrt[3]{d},\zeta_3)$ whose $3$-class group is of type $(9,3)$

Abstract

Let $\mathrm{k}=\mathbb{Q}(\sqrt[3]{d},\zeta_3)$, with $d$ a cube-free positive integer. Let $C_{\mathrm{k},3}$ be the $3$-component of the class group of $\mathrm{k}$. By the aid of genus theory, arithmetic proprieties of the pure cubic field $\mathbb{Q}(\sqrt[3]{d})$ and some results on the $3$-class group $C_{\mathrm{k},3}$, we are moving towards the determination of all integers $d$ such that $C_{\mathrm{k},3} \simeq \mathbb{Z}/9\mathbb{Z}\times \mathbb{Z}/3\mathbb{Z}$.