Surjective morphism of complete non-singular curves is normalization

manpreet Best Answer 3 years ago

My syllabus on algebraic geometry states the following:

''Let ϕ:X→Yϕ:X→Y be a surjective morphism of complete non-singular curves. Then XX is the normalization of YY in the function field of XX. In particular, the morphism ϕϕ is finite. ''

How do we know that XX is the normalization of YY?

The definition of normalization we use is the following:

''A normalization of YY in k(X)k(X) is a finite surjective morphism π:X→Yπ:X→Y such that XX is normal and the extension of function fields corresponding to ππ is the extension k(Y)⊃k(X)k(Y)⊃k(X). ''

I understand that ϕϕ is surjective and XX is normal (ie. OX,POX,P is integrally closed ∀P∈X∀P∈X) because it is non-singular, but how do we know that the extension of function fields is exactly