I need to prove that if $F$ is a field and $u=\frac{f(t)}{g(t)} \in F(t)$ (where $f,g$ are coprime in $F[t]$) then $[F(t):F(u)]=\max(\deg f,\deg g)$.
I know I have to prove that $ug(x)-f(x)$ is irreducible in $F(u)[x]$, but I cannot do it. Any hint will be appreciated.