I am trying to do this exercise of Fulton's Algebraic Curves:
6.45 Let $C, C'$ be curves, $F$ a rational map from $C'$ to $C$. Prove: (a) Either $F$ is dominating, or $F$ is constant (i.e., for some $P\in C$, $F(Q)=P$, all $Q\in C'$). (b) If $F$ is dominating, then $k(C')$ is a finite algebraic extension of $\hat{F}(k(C))$.
To conclude the problem i have to show the following:
If $E/F$ is an extension of fields over $k$, such that $E$ and $F$ have the same transcendental degree(in this case 1) then $E/F$ is algebraic.
I would like a hint to prove that. Thank you.