I understand that holomorphic square roots of the canonical line bundle of a compact Riemann surface always exist, and that there are $2^{2g}$ choices of such a root.
But what about further roots? When the genus of the Riemann surface is larger than $1$, we can ask for further roots, such as a square root of a square root of the canonical bundle.
Is this always possible? In other words, if $K$ is the canonical bundle, can I always keep taking roots so that eventually I have $K=L^{2g-2}$ for some degree $1$ holomorphic line bundle $L$?
Why or why not?