Prove that for a Riemann surface of non-zero genus, $p - q$ is not a principal divisor for any two distinct points $p$ and $q$.
My idea is to use Riemann-Roch. Suppose $D=p-q$ is a prinipal divisor, then $\deg(D)=0$. Then I need to show $h^0(\mathcal{O} (p-q))-h^1(\mathcal{O}(p-q))=1$. But I don't know how to proceed. Is this method correct? Thanks.
I think the answer you're looking for is much simpler. If $p - q$ is a principal divisor, then there exists a meromorphic function $f$ on your Riemann surface $X$ with a unique (single) zero at $p$ and a unique (single) pole at $q$. This function $f$ then defines a degree one map from $X$ to $\mathbb P^1$. Therefore, $X$ has genus zero, by Riemann-Hurwitz.