Determine the group $\text{Aut}(\Bbb Q(\sqrt[3]{2}))$.
$\Bbb Q(\sqrt[3]{2})$ is the splitting field of the polynomial $x^3-2$ over $\Bbb Q$. So $$\Bbb Q(\sqrt[3]{2})=\{a+b\sqrt[3]2+c\sqrt[3]{2}^2 \mid a,b,c \in \Bbb Q\}$$
If $\varphi:\Bbb Q(\sqrt[3]{2}) \to \Bbb Q(\sqrt[3]{2})$ is an automorphism, then $$\varphi(a+b\sqrt[3]2+c\sqrt[3]{2}^2)=a+b\varphi(\sqrt[3]{2})+c\varphi(\sqrt[3]{2})^2$$ So $\varphi$ is determined by where it sends $\sqrt[3]{2}$.
How can I find out where $\varphi$ sends $\sqrt[3]{2}$ in order to see how the elements of $\text{Aut}(\Bbb Q(\sqrt[3]{2}))$ should behave?