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 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

X^p - X + \alpha,\,

for $\alpha$ in K, is called an Artin–Schreier polynomial. 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 its splitting field over K is a cyclic extension of K of degree p. This follows since for any root β, the numbers β + i, for $1\leq i\leq p$ , form all the roots—by Fermat's little theorem—so the splitting field is $K(\beta)$ .

Conversely, any Galois extension of K of degree p 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. These extensions are called Artin–Schreier extensions.

Artin–Schreier 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 Artin–Schreier extension or a purely inseparable extension.

Artin–Schreier–Witt extensions

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).

References

Artin, Emil; Schreier, Otto (1927), "Eine Kennzeichnung der reell abgeschlossenen Körper", Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, Springer Berlin / Heidelberg, 5: 225–231, doi:10.1007/BF02952522, ISSN 0025-5858
Lang, Serge (2002), Algebra, Graduate Texts in Mathematics, 211 (Revised third ed.), New York: Springer-Verlag, ISBN 978-0-387-95385-4, Zbl 0984.00001, MR 1878556 Section VI.6
Neukirch, Jürgen; Schmidt, Alexander; Wingberg, Kay (2000), Cohomology of Number Fields, Grundlehren der Mathematischen Wissenschaften, 323, Berlin: Springer-Verlag, ISBN 978-3-540-66671-4, Zbl 0948.11001, MR 1737196 Section VI.1
Witt, Ernst (1936), "Zyklische Körper und Algebren der Characteristik p vom Grad pⁿ. Struktur diskret bewerteter perfekter Körper mit vollkommenem Restklassenkörper der Charakteristik pⁿ", Journal für die reine und angewandte Mathematik (in German), 176: 126–140, doi:10.1515/crll.1937.176.126

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