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.

connectedRiemann surfaces, to avoid silly falsities about details. – paul garrett Jul 15 '11 at 23:51