$F(f)\subset F(x)$ is an algebraic extension of degree $\max(p,q)$ if $f$ is non constant quotient polynomial

Question

Let $F$ be a field, with $f=a/b\in F(x)$ the quotient of coprime polynomials $a,b\in F[x]$ of degree $p$ and $q$ respectively.

How do I prove that if $f$ is not constant, then $F(f)\subset F(x)$ is an algebraic extension of degree $\max(p,q)$?

I honestly have no idea how to approach this problem, maybe someone can give me a push in the right direction?

Answer 1

Let $f(x) = t$. Then you can think of your extension as $F(t) \subset F(t)[x]/(a(x)-tb(x))$. I can elaborate more but you should try finishing the proof from here on.