# Compact Riemann surfaces and Algebraic Functions

Good evening,

In Riemann surfaces by Otto Forster there is the following theorem : Let $X$ be a Riemann surface and $P(T)=T^n+c_1T^{n_1}+\ldots + c_n\in\mathcal{M}(X)[T]$ an irreducible polynomial of degree $n,$ where $\mathcal{M}(X)$ is the set of all meromorphic functions on $X.$ Then there exist a Riemann surface $Y,$ a branched holomorphic n-sheeted covering $\pi : Y\to X$ and a meromorphic function $F$ on $Y$ such that $(\pi^{\ast}P)(F) = 0$.

We call $Y$ the algebraic function defined by the polynomial $P(T).$ (I don't restate the uniqueness of this Riemann surface).

My question : If $X$ is a compact Riemann surface, can we consider it as an algebraic function defined by some irreducible polynomial $P(T)\in\mathcal{M}(\mathbb{P}^1)[T]$?

I'm thinking of meromorphic functions on $X,$ which we can consider them as holomorphic mappings $X\to\mathbb{P}^1,$ having the smallest positive degree,i.e the inverse image of each point of $\mathbb{P}^1$ contains the smallest number of points. But I'm not sure.

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Yes. That's the Riemann existence theorem, every compact Riemann surface is an algebraic curve over $\mathbb{C}$. – Felipe Voloch Jul 15 '11 at 23:34
Just to quibble, maybe it's best to talk about compact connected Riemann surfaces, to avoid silly falsities about details. – paul garrett Jul 15 '11 at 23:51
Thank you very much for this information. I will search for this theorem. – Đức Anh Jul 15 '11 at 23:55
@ paul garret : Yes, i'm sorry. In Otto Forster's book, he has a convention that every Riemann surface is connected. – Đức Anh Jul 15 '11 at 23:56
BTW, the Riemann existence is in Forster (Cor. 14.13 in my edition). – Torsten Ekedahl Jul 16 '11 at 5:09