Find degree of extension $[\mathbb{Q}(\sqrt[3]{2},\sqrt{2}):\mathbb{Q}(\sqrt[3]{2})]$.
My approach was the following:
Consider the polynomial $x^2-2\in \mathbb{Q}[x]$ and $\sqrt{2}$ is its root;
this shows that $[\mathbb{Q}(\sqrt[3]{2},\sqrt{2}):\mathbb{Q}(\sqrt[3]{2})]\leq 2$.
I think that this equals $2$, but I cannot prove it rigorously.
I would be very grateful if anyone can show how to solve this problem.
BTW, please do not use any Galois theory, because I am not familiar with it yet.