So the title gives the jist of my question. Specifically, let $X$ be a non-singular projective curve, $P$ a point on $X$, $v_P$ the discrete valuation associated to the ring $\mathcal{O}_P$. Then I have read that the completion of $k(X)$ with respect to the valuation $v_P$ is isomorphic to the field of formal Laurent series over $k$.
Stuff that might be relevant? I know some basic Galois theory, some very basic point set topology, and I'm just starting chapter 10 in An Introduction to Commutative Algebra by Atiyah and MacDonald.
I was hoping someone could tell me the material I will have to read to understand this along with good books that cover it. If there is an algebraic way of going about this I would prefer it as I'm really enjoying An Introduction to Commutative Algebra. Also if someone wanted to give me an overview of what is happening here that would be appreciated also.
Thanks for any help!