Let $\zeta_n$ be the $n^{th}$ root of unity $\zeta_n=e^{2\pi i/n}$. How can I prove that $\zeta_5\notin \mathbb{Q}(\zeta_7)$?

Question

This question is from Artin 15.3.3: Let $\zeta_n$ be the $n^{th}$ root of unity $\zeta_n=e^{2\pi i/n}$. How can I prove that $\zeta_5\notin \mathbb{Q}(\zeta_7)$? I'm quite stuck so any help would be much appreciated!

Answer 1

Hint: What is the degree $[\mathbb{Q}(\zeta_7) : \mathbb{Q}]$? What is the degree $[\mathbb{Q}(\zeta_5) : \mathbb{Q}]$?