This is related to Forster Riemann surface Ex 8.1
Let $X,Y$ be two compact riemann surfaces. Let $M(X)$ and $M(Y)$ be meromorphic function fields of $X$ and $Y$ respectively. If $M(X)\cong M(Y)$, then $X\cong Y$ holomorphically.
$Q1:$Hint says represent $X$ and $Y$ by algebraic functions. I do not have $GAGA$ at disposal. If I can use $GAGA$, then I am done as I obtain birational map first and extend by singularity having codimension less than 2. How do I represent $X,Y$ by algebraic functions? Since the book does not assume algebraic geometry, I guess there should be a way without resorting to algebraic geometric argument.
$Q2:$ I tried the following take any $f\in M(X)$ and its image $f'\in M(Y)$. I can construct algebraic surface $\tilde{X},\tilde{Y}$covering $X,Y$ respectively. Then I can replace $\tilde{Y}$ by $\tilde{X}$ via direct construction through holomorphic sheaf's etale espace of $Y$ and $X$ and matching the fiber to identify. This process can be reversed. Thus $\tilde{Y}\cong\tilde{X}$. I want to descend this biholomorphism down to $X\to Y$. How should I proceed?