2 Filled out the definition, as it was incomplete

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 $$y^2 - g(xh(x) y = f(x),\ f(x),g(x)\in f(x),h(x)\in k[x]. ,$$ 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

1

# Hyperelliptic curves over characteristic two fields

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 $$y^2 - g(x) y = f(x),\ f(x),g(x)\in k[x].$$ 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.