Artin–Schreier curve
In mathematics, an Artin–Schreier curve is a plane curve defined over an algebraically closed field of characteristic formula_1 by an equation for some rational function formula_3 over that field. One of the most important examples of such curves is hyperelliptic curves in characteristic 2, whose Jacobian varieties have been suggested for use in cryptography. It is common to write these curves in the form for some polynomials formula_3 and formula_6. Definition. More generally, an "Artin-Schreier curve" defined over an algebraically closed field of characteristic formula_1 is a branched covering of the projective line of degree  formula_1. Such a cover is necessarily cyclic, that is, the Galois group of the corresponding algebraic function field extension is the cyclic group formula_10. In other words, formula_11 is an Artin–Schreier extension. The fundamental theorem of Artin–Schreier theory implies that such a curve defined over a field formula_12 has an affine model for some rational function formula_14 that is not equal for formula_15 for any other rational function formula_16. In other words, if we define polynomial formula_17, then we require that formula_18. Ramification. Let formula_19 be an Artin–Schreier curve. Rational function formula_3 over an algebraically closed field  formula_12 has partial fraction decomposition for some finite set formula_23 of elements of formula_12 and corresponding non-constant polynomials formula_25 defined over formula_12, and (possibly constant) polynomial formula_27. After a change of coordinates, formula_3 can be chosen so that the above polynomials have degrees coprime to formula_1, and the same either holds for formula_27 or it is zero. If that is the case, we define Then the set formula_32 is precisely the set of branch points of the covering formula_8. For example, Artin–Schreier curve formula_2, where formula_3 is a polynomial, is ramified at a single point over the projective  line. Since the degree of the cover is a prime number, over each branching point formula_36 lies a single ramification point formula_37 with corresponding ramification index equal to Genus. Since formula_1 does not divide formula_40, ramification indices formula_41 are not divisible by formula_1 either. Therefore, the Riemann–Roch theorem may be used to compute that the genus of an Artin–Schreier curve is given by For example, for a hyperelliptic curve defined over a field of characteristic formula_44 by equation formula_45 with formula_3 decomposed as above, Generalizations. "Artin–Schreier" curves are a particular case of plane curves defined over an  algebraically closed field formula_12 of characteristic formula_1 by an equation for some separable polynomial formula_51 and rational function formula_18. Mapping formula_53 yields a covering map from the curve formula_54 to the projective line formula_55. Separability of defining polynomial formula_56 ensures separability of the corresponding function field extension formula_11. If formula_58, a change of variables can be found so that formula_59 and formula_60. It has been shown that such curves can be built via a sequence of Artin-Schreier extension, that is, there exists a sequence of cyclic coverings of curves each of degree formula_1, starting with the projective line. Artin