My syllabus on algebraic geometry states the following:
''Let $\phi: X \to Y$ be a surjective morphism of complete non-singular curves. Then $X$ is the normalization of $Y$ in the function field of $X$. In particular, the morphism $\phi$ is finite. ''
How do we know that $X$ is the normalization of $Y$?
The definition of normalization we use is the following:
''A normalization of $Y$ in $k(X)$ is a finite surjective morphism $\pi: X \to Y$ such that $X$ is normal and the extension of function fields corresponding to $\pi$ is the extension $k(Y) \supset k(X)$. ''
I understand that $\phi$ is surjective and $X$ is normal (ie. $\mathcal{O}_{X, P}$ is integrally closed $\forall P \in X$) because it is non-singular, but how do we know that the extension of function fields is exactly $k(Y) \supset k(X)$ and that $\phi$ is a finite morphism?