I want to show the assertion above.
I already know: $p$ splits completely in $\mathbb{Q}(\zeta\_3,\sqrt[3]{2}) \Leftrightarrow p\mathcal{O}\_{\mathbb{Q}(\zeta\_3)}=R \overline{R}$ in the ring of integers $\mathcal{O}\_{\mathbb{Q}(\zeta\_3)}=\mathbb{Z}[\zeta\_3]$ and $R$ is a principal ideal, i.e. $R=(\pi)$ where $\pi\in \mathbb{Z}[\zeta\_3]$, and $6\mid (\pi-1)$. (This equivalence uses class fields theory)
It is claimed that the second one is equivalent to: $p\mathcal{O}\_{\mathbb{Q}(\zeta\_3)}=(\pi)(\overline{\pi})$ and $\pi=x+3y\sqrt{-3}$ .
Why is it true?