Artin-Schreier theory

:"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 $alpha$ in "K", is called an "Artin-Schreier polynomial". It can be shown that when $alpha$ 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.