I'm having some trouble approaching the following problem:
Let $m, n \in \mathbb{Z}^+$ be such that $(m,n)=1$. Let $\alpha$ be a primitive $m$-th root of unity and $\beta$ be a primitive $n$-th root of unity. Show that $\mathbb{Q}(\alpha)\cap\mathbb{Q}(\beta)=\mathbb{Q}$.
I've tried computing the irreducible polynomial of $\alpha$ over $\mathbb{Q}(\beta)$ to show that it has degree $\varphi(m)$. Does this way lead to the desired result?