Put $z_{n} = e^{2\pi i /n}$. I am searching for $n \in \mathbf{N}$ so that $\mathbf{Q}(z_{5},z_{7}) = \mathbf{Q}(z_{n})$.
I know that : $z_{5} = \cos(\frac{2\pi}{5})+i\sin(\frac{2\pi}{5}) $ and $z_{7} =\cos(\frac{2\pi}{7})+i\sin(\frac{2\pi}{7})$.
Can you give me a hint how to continue my search? Thank you.