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

for α in K, is called an Artin–Schreier polynomial. It can be shown that when α does not lie in the subset , 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.^{[citation needed]}

[edit] References

• Section VI.6 of Lang, Serge (2002), Algebra, Graduate Texts in Mathematics, 211 (Revised third ed.), New York: Springer-Verlag, MR1878556, ISBN 978-0-387-95385-4 

• Section VI.1 of Neukirch, Jürgen; Schmidt, Alexander; Wingberg, Kay (2000), Cohomology of Number Fields, Grundlehren der Mathematischen Wissenschaften, 323, Berlin: Springer-Verlag, MR1737196, ISBN 978-3-540-66671-4