# Artin–Schreier curve

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In mathematics, an **Artin–Schreier curve** is a plane curve defined over an algebraically closed field of characteristic by an equation

for some rational function over that field.

## Definition

More generally, an *Artin-Schreier curve* defined over an algebraically closed field of characteristic is a branched covering

of the projective line of degree . Such a cover is necessarily cyclic, that is, the Galois group of the corresponding algebraic function field extension is the cyclic group . In other words, is an Artin–Schreier extension.

The fundamental theorem of Artin–Schreier theory implies that such a curve defined over a field has an affine model

for some rational function that is not equal for for any other rational function . In other words, if we define polynomial , then we require that .

## Ramification

Let be an Artin–Schreier curve. Rational function over an algebraically closed field has partial fraction decomposition

for some finite set of elements of and corresponding non-constant polynomials defined over , and (possibly constant) polynomial . After a change of coordinates, can be chosen so that the above polynomials have degrees coprime to , and the same either holds for or it is zero. If that is the case, we define

Then the set is precisely the set of branch points of the covering .

For example, Artin–Schreier curve , where is a polynomial, is ramified at a single point over the projective line.

## Generalizations

*Artin–Schreier* curves are a particular case of plane curves defined over an algebraically closed field of characteristic by an equation

for some separable polynomial and rational function . Mapping yields a covering map from the curve to the projective line . Separability of defining polynomial ensures separability of the corresponding function field extension . If , a change of variables can be found so that and {{ safesubst:#invoke:Unsubst||date=__DATE__ |$B=
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