I've never seen written in a book a way or an algorithm for computing Riemann surfaces of a given algebraic function. I would like to know how to construct such Riemann surface using intuitive cutting and pasting techniques as people used to do long time ago.
In general, the general statement is that given $Y$ a Riemann surface and some polynomial $P(T) \in \mathscr{M}(Y)[T]$ of degree $n$, one can find another Riemann surface $X$ such that $\pi: X \longrightarrow Y$ is an holomorphic $n$-branched covering and a meromorphic function $F \in \mathscr{M}(X)$ such that $\pi^{*}(P(F)) =0$. So, in summary, given a Riemann surface and a multivalued function one can always find another Riemann surface such that the multivalued function is a meromorphic function.
However the proof of the statement above, in general, is done by constructing $X$ with the sheaf of holomorphic functions (as an étale space) and extending germs (to get the "monodromy" information). Therefore, the proof does not show a general way of computing such surfaces.
Furthermore, I would be glad if someone could show a good example of such construction.
Thanks in advance.