I'm a bit lost with the general form of holomorphic one forms on the compact hyperelliptic Riemann surface $M$ defined by $w^2=h(z)$ where $h$ can have an even or odd number of distinct, simple zeroes.
I need to find that $\frac{dz}{w}$ is holomorphic if the genus $g$ of $M$ is at least $1$ and as a follow up that for all polynomials $p(z)$ with $deg(p)<g, p(z)\frac{dz}{w}$ is a holomorphic differential as well.
For the first part I just considered where $\frac{dz}{w}$ could have poles, which would be the branch points. But locally, $w$ goes as $\sqrt z$ so this does seem to indicate a pole to me, or can I not use a kind of two-sheet model for this? I see people saying that this is a trivial fact so maybe I'm going about it the wrong way.
Any hints would be appreciated, if at all possible I would still try to find it myself. Thanks!