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.
> 
> 
> 


I don't see what was wrong with my initial intuition, so if anyone could tell me, or explain why the definition given is correct, I would be much obliged.


edit - to give fuller definition



Hyperelliptic curves over characteristic two fields
If anyone is interested, a thorough treatment is given by the book referenced in the comments here: <https://math.stackexchange.com/questions/137495/artin-schreier-extensions-over-characteristic-two-fields>


