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Let $\mathrm{k}=\mathbb{Q}(\sqrt[3]{d},\zeta_3)$, with $d$ a cube-freepositive integer. Let $C_{\mathrm{k},3}$ be the $3$-component of the classgroup of $\mathrm{k}$. By the aid of genus theory, arithmetic proprieties ofthe 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 ofall integers $d$ such that $C_{\mathrm{k},3} \simeq\mathbb{Z}/9\mathbb{Z}\times \mathbb{Z}/3\mathbb{Z}$.

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