I am asked to prove that "$\sqrt[3]{2}\not\in\mathbb{Q}(\alpha_n)$ for all $n$ where $\alpha_n$ is the primitive nth root of unity"
I have attempted to use contradiction with the tower theorem. I got stuck at $[\mathbb{Q}(\alpha_n):\mathbb{Q}(\sqrt[3]{2})]=\frac{\phi(n)}{3}$ where $\phi(n)$ is the Euler totient function. Can someone give me some sort of hint?
Thanks in advance!