I am reading the paper An algorithm for computing the Weierstrass normal form. I have a question about the function field of an algebraic curve.
Let $f(x,y)$ be a polynomial in $x,y$. Let $L$ be the field generated by the coefficients of $f$. Let $C$ be the curve given by $f(x,y)=0$.
On page 2, Section 2, it is said that the function field $\overline{L}(C)$ can be identified with $\overline{L}(x)[y]/(f)$.
How to prove this fact? Thank you very much.