Why are these extension fields equal?

Question

I have $\alpha= 2^{1/3}+\sqrt{2}$ and $E(\alpha)=\mathbb{Q}$, I want to prove that $E=\mathbb{Q}(2^{1/3}, \sqrt{2})$.

I tried to compute the irreducible polynomial of $\alpha$, but I didn't get anywhere. Any hint?