I have been looking at hyperelliptic curves over an algebraically closed field $k$ of characteristic two, with a view towards finding the basis for the vector space of holomorphic differentials. To do this I have viewed the curves as the function field $k(x,y)$, originally restricting to those defined by $ y^2 - y = f(x), \ f(x)\in k[x]. $
After this, I thought that to generalise to all hyperelliptic curves I should allow $f(x)$ to be any rational function. However, looking at the literature it seems like the definition is instead:
A hyperelliptic curve of genus $g$ ($g\geq 1$) is an equation of the form $ y^2 - h(x) y = f(x),\ f(x),h(x)\in k[x], $ where the degree of $h(x)$ is at most $g$, and $f(u)$ is a monic polynomial of degree $2g +1$, with no elements of $k\times k$ satisfying the original equation and both of it's partial derivatives.
Given a function of the above form, we can divide by $h^2$ and then if we let $y' = \frac{y}{h}$ we have $y'^2 - y' = \frac{f(x)}{h(x)}$, so it does correspond to an Artin-Schreier extension with a rational function on the right hand side. However, the other direction is not so clear to me.
Any idea as to the best literature to look at Artin-Schreier curves over finite characteristic would be almost as good as a direct answer.