At the moment, I try to understand the connection between an non-singular projective algebraic curve and an algebraic function field. The field $K$ is algebraically closed.
If an algebraic curve $\mathcal{C}$ is non-singular, then there exists a isomorphism between the points $P\in\mathcal{C}$ and the places of the function field $K(\mathcal{C})$ given by$$ P\mapsto\mathcal{M}_P $$where $\mathcal{M}_P$ is the maximal ideal of the local ring $\mathcal{O}_P$. Define $F_P=\mathcal{O}_P/\mathcal{M}_P$. Then the following holds:$$ [F_P:K]=1. $$according to Stichtenoth's "Algebraic function fields and codes" (It should be noted I deduce this fact by the appendix and comparing definitions). But why? I am not a very smart person, but does this follow from the weak version of Hilbert's Nullstellensatz? If I can show that $F_P$ is a finitely-generated $K$-algebra, then I know that $F_P\cong K$, therefore I get the desired result.
I am grateful for any help!
Sincerely, Hypertrooper