# Artin-Schreier extensions 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:

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.

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What do you mean by "the other direction" ? – user18119 May 12 '12 at 16:52
Well, what I described goes from the definition that is highlighted to the general definition. If they are equivalent, then I should be able to take something in the form $y^2- h(x)y = f(x)$ for polynomials $h(x)$ and $f(x)$, and find rewrite it as $y^2 - y = a(x)$ for a rational function $a(x)$, i.e. such that the extension by $y$ is the same. Is that clear? Edit: To clarify, by general definition, I mean the extension defined by $y^2 - y = f(x)$ for a rational function $f(x)$. – Joe Tait May 12 '12 at 23:43
OK. You can find a proof of the equivalence between "highlighted" and "general" definitions in "Algebraic geometry and arithmetic curves", Prop 7.4.24. – user18119 May 13 '12 at 16:55
That is brilliant, thank you very much. Looks like a good book in general too :) – Joe Tait May 13 '12 at 18:18
You might have a look at the book "Algebraic Function Fields and Codes" by Henning Stichtenoth. – user18119 May 14 '12 at 20:21