In mathematics, ArtinâSchreier 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 ArtinâSchreier 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 in K, is called an ArtinâSchreier polynomial. When for all , this polynomial is irreducible in KX], and its splitting field over K is a cyclic extension of K of degree p. This follows since for any root ÎČ, the numbers ÎČ + i, for , form all the rootsâby Fermat's little theoremâso the splitting field is .
Conversely, any Galois extension of K of degree p equal to the characteristic of K is the splitting field of an ArtinâSchreier polynomial. This can be proved using additive counterparts of the methods involved in Kummer theory, such as Hilbert's theorem 90 and additive Galois cohomology. These extensions are called ArtinâSchreier extensions.
ArtinâSchreier extensions play a role in the theory of solvability by radicals, in characteristic p, representing one of the possible classes of extensions in a solvable chain.
They also play a part in the theory of abelian varieties and their isogenies. In characteristic p, an isogeny of degree p of abelian varieties must, for their function fields, give either an ArtinâSchreier extension or a purely inseparable extension.
There is an analogue of ArtinâSchreier theory which describes cyclic extensions in characteristic p of p-power degree (not just degree p itself), using Witt vectors, developed by Witt ( 1936).