In mathematics, Artin–Schreier theory is a branch of Galois theory, specifically a positive characteristic analogue of Kummer theory, for Galois extensions of degree equal to the characteristic p. Artin and Schreier (1927) introduced Artin–Schreier theory for extensions of prime degree p, and Witt (1936) generalized it to extensions of prime power degree pn.
If K is a field of characteristic p, a prime number, any polynomial of the form for
Conversely, any Galois extension of K of degree p equal to the characteristic of K is the splitting field of an Artin–Schreier polynomial.
They also play a part in the theory of abelian varieties and their isogenies.
There is an analogue of Artin–Schreier theory which describes cyclic extensions in characteristic p of p-power degree (not just degree p itself), using Witt vectors, developed by Witt (1936).