Divisibility between the degree of two extension fields

Question

I am working on this problem:

Let $K$ be an extension field of a field $F$, and let $\alpha \in K$ be algebraic over $F$, with minimal polynomial $p(x)$. Let $\beta \in F(\alpha)$ be algebraic over $F$, with minimal polynomial $q(x)$. Prove that $\deg(q)\mid \deg(p)$.

I'm stuck at starting this problem so I would appreciate some hints for it.

Answer 1

Hint:

$$\dim_F F(\alpha)=\dim_{F(\beta)} F(\alpha)\cdot \dim_F F(\beta).$$