prove that $\{1,\sqrt[3]{3},(\sqrt[3]{3})^2 \}$ is integral basis for $\Bbb{Q}(\sqrt{3^3})$
my aim is to show that $\Bbb{Q}_K=\Bbb{Z}+\sqrt[3]{3}\Bbb{Z}+(\sqrt[3]{3})^2\Bbb{Z}$
$\supseteq$ : it is clear
$\subseteq$ : let $\alpha \in \Bbb{Q}_K$ i want to show $\alpha=z_1+z_2\sqrt[3] {3}+z_3(\sqrt[3]{3})^2$ and $z_1,z_2,z_3 \in \Bbb{Z}$
is there any short way to show this ? Thanks a lot.