Hi,

I need some help with the following exercise.

Let $\displaystyle L= \mathbb{Q}(\sqrt[3]{2})$. Show that $\displaystyle [L:\mathbb{Q}]=3$ and that $\displaystyle L$ has no automorphisms other than the identity (hint: $\displaystyle T^3-2$ has no root in $\displaystyle \mathbb{Q}$ and a single root in $\displaystyle \mathbb{R}$).

I have no idea how I can prove $\displaystyle \dim_{L}\mathbb{Q}=3$ and also no idea for the automorphism.

Can someone give me a hint?

Thanks in advance!