Residue field of the integral closure of a local ring in its field of fractions When considering the discrete valuation rings contained in the rational functions field $R(F)$ of an irreducible plane projective curve $F \in \mathbb{P}^2(K)$ ($K$ algebraically closed), one can find that these are of the form $(\overline{\cal{O}_{F,P}})_{\cal{M}}$, i.e the localisation of the integral closure of $\cal{O}_{F,P}$ in $R(F)$ at one of its finitely many maximal ideals (where $\cal{O}_{F,P}=\cal{O}_P/(F)$, with $\cal{O}_P$ the rational functions defined at $P$).
How can I show that the residue fields of all these rings is then always $K$ ?
P.S I'm using Ernst Kunz's Introduction to plane algebraic curves, Chapter 6 as reference.
 A: For your statement to be true you should assume that these discrete valuation rings contain your field $K$.
You may assume that $F$ is a normal curve. You know that for a closed point $p\in F$ there exists an affine open neighbourhood $U\subseteq F$ of $p$. Write $U=\mathrm{Spec}(A)$ for some finitely generated $K$-algebra. Now $p$ corresponds to some maximal ideal $\mathfrak{m}$ of $A$ and the residue field corresponds to $A/\mathfrak{m}$. By some standard results in commutative algebra $A/\mathfrak{m}$ is a finite extension of $K$. Hence it is isomorphic with $K$.
A: Found it ! Thanks to Slup for making me realize I had overlooked the hypothesis that $K$ was a subset of the considered valuation rings.
Actually, since $K \subset \cal{O}_{F,P}$, $(\overline{\cal{O}_{F,P}})_{\cal{M}} \subset R(F)=Q(\cal{O}_{F,P})$ and the residue field of $\cal{O}_{F,P}$ is $K$, an element of $(\overline{\cal{O}_{F,P}})_{\cal{M}}$ writes $\frac{x}{y}$ with $x$, $y$ in $\cal{O}_{F,P}$. These elements are both equivalent modulo the maximal ideal of $\cal{O}_{F,P}$ to some element of $K$ so $\frac{x}{y}$ is equivalent to an element of $K\subset \cal{O}_{F,P}$ modulo the maximal ideal of $(\overline{\cal{O}_{F,P}})_{\cal{M}}$. This shows that the canonical injection from the residue field of $\cal{O}_{F,P}$ to the residue field of $(\overline{\cal{O}_{F,P}})_{\cal{M}}$ is also surjective.
