I have a question stemming from a line in Hartshorne's Algebraic Geometry, Section 1.6. I'll put the quote here:
If $P$ is a point on a nonsingular curve $Y$, then by (5.1) the local ring $\mathcal{O}_p$ is a regular local ring of dimension one, and so by (6.2A) is is a discrete valuation ring. Its quotient field is the function field $K$ of $Y$, and since $k \subseteq \mathcal{O}_p$, it is a valuation ring of $K/k$.
My only question is how does the last line follow? Is it a general fact that a valuation of a quotient field $K$ obtained from an integral domain $R$ vanishes on any subfield of $R$? If so, where can I see a proof of that?
I've tried playing around with this and seem to be stuck, although I'm almost positive that the answer is right in front of me.