Prove that $e^{2\pi i/5}$ is not in the $7$-th cyclotomic field.

Question

Let $\xi_n = e^\frac{2\pi i}{5} $. Prove that $\xi_5 \notin \Bbb{Q}(\xi_7)$ where $\Bbb{Q}(\xi_7)$ is the 7-th cyclotomic field.

How would I approach this question? I'm having a difficult time coming up with a solution, any tips would be appreciated

Answer 1

Suppose $\;\xi_5\in\Bbb Q(\xi_7)\;$, then

$$\;\phi(5)=4=[\Bbb Q(\xi_5):\Bbb Q]\,\mid\,[\Bbb Q(\xi_7):\Bbb Q]=6=\phi(7)\;$$

and this is absurd.