Artin-Schreier theory

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See Artin-Schreier theorem for theory about real-closed fields.

In mathematics, Artin-Schreier theory is a branch of Galois theory, and more specifically is a positive characteristic analogue of Kummer theory, for extensions of degree equal to the characteristic p.

If K is a field of characteristic p, a prime number, any polynomial of the form

X^p - X + \alpha,\,

for α in K, is called an Artin-Schreier polynomial. It can be shown that when α does not lie in the subset \{ y \in K \, | \, y=x^p-x \; \mbox{for } x \in K \}, this polynomial is irreducible in K[X], and that its splitting field over K is a cyclic extension of K of degree p. The point is that for any root β, the number β + 1 is again a root.

Conversely, any Galois extension of K of degree p (remember, p is equal to the characteristic of K) is the splitting field of an Artin-Schreier 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.

Artin-Schreier extensions, as are called those arising from Artin-Schreier polynomials, 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 Artin-Schreier extension or a purely inseparable extension.

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, which were developed by Witt for precisely this reason.

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