Let $K$ be a field, $p, q ∈ K[X]$ coprime with deg $p = m$, deg $q = n$, and suppose $f := \frac{p}{q} ∈ K(X)\setminus K$.
Supposedly, then, $K(X)/K(f)$ is an algebraic extension of degree max$\{m,n\}$. To show this we could perhaps invoke the Fundamental Theorem of Galois Theory. Namely, if $K(f) = K(X)^G$ for some finite $G ≤\text{Aut}(K(X))$, then the Fundamental Theorem states that $[K(X) :K(f)] = \#G$, which is finite. (Therefore the extension is immediately algebraic.) This group would obviously depend on $p, q$ and hence on $m, n$, so we could write $G_{m,n}$. But what group has order max$\{m, n\}$??