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| Artin–Schreier theory | |
|---|---|
| Name | Artin–Schreier theory |
| Field | Algebraic number theory, Algebra |
| Introduced | 1927 |
| Contributors | Emil Artin, Otto Schreier |
Artin–Schreier theory is a framework in algebraic number theory and algebra that classifies degree-p extensions of fields of characteristic p, linking field theory, Évariste Galois-style Galois theory, and cohomology. Developed by Emil Artin and Otto Schreier, the theory connects explicit polynomial equations to group-theoretic and cohomological invariants used by mathematicians working in contexts influenced by David Hilbert, Helmut Hasse, and Claude Chevalley.
Artin–Schreier theory addresses cyclic extensions of prime degree p for fields of characteristic p, situating itself alongside Kummer theory, Galois theory, and the innovations of Noether, Artin, and Tate. It replaces multiplicative norm conditions in Kummer theory by additive conditions tied to the Frobenius endomorphism central to the work of Émile Picard and André Weil. The theory has been used in investigations by researchers such as Serge Lang, John Tate, and Alexander Grothendieck.
An Artin–Schreier extension is a cyclic extension of degree p of a field F of characteristic p, analogous to degree-n cyclic extensions in Kummer theory over fields containing n-th roots of unity considered by Ernst Eduard Kummer and Richard Dedekind. Such extensions often arise in the study of function fields over Finite fields like Évariste Galois’s original examples and in work by Hasse on local fields. Constructions of these extensions parallel techniques used by Carl Friedrich Gauss in cyclotomy and by Kronecker in class field contexts later developed by Heinrich Weber and Hilbert.
Artin–Schreier polynomials take the form X^p − X − a over a field F of characteristic p, echoing the polynomial families studied by Évariste Galois and later by Emil Artin. The equation X^p − X = a determines an extension F(α)/F when α is a root, an approach reminiscent of constructions in Galois theory used by Niels Henrik Abel and applied in explicit examples by Camille Jordan and Otto Schreier. Irreducibility criteria for these polynomials relate to work on separability and inseparability investigated by Oscar Zariski and Pierre Deligne.
The Galois group of an Artin–Schreier extension is cyclic of order p, a fact that connects to classical results of Évariste Galois and modern formulations by Emil Artin and John Tate. The set of Artin–Schreier extensions of F corresponds to a quotient of the additive group F by the image of the Frobenius map x ↦ x^p − x, an identification akin to the ideals-to-extensions correspondence in the Artin reciprocity framework championed by Helmut Hasse and Teiji Takagi. This group-theoretic viewpoint parallels the role of cyclotomic fields in Kummer theory explored by Leopold Kronecker.
Cohomologically, Artin–Schreier theory appears through H^1 of the absolute Galois group with coefficients in the additive group, aligning with the methods of Claude Chevalley and Jean-Pierre Serre. The short exact sequence defined by the Frobenius endomorphism gives rise to a connecting homomorphism in Galois cohomology, mirroring constructions by Alexander Grothendieck in étale cohomology and later by Grothendieck and Jean-Louis Verdier in derived contexts. These interpretations integrate with duality theorems such as those developed by John Tate.
Artin–Schreier extensions provide explicit examples in the theory of function fields over finite fields like those studied by André Weil and applied in coding theory influenced by Richard E. Blahut and Vladimir Levenshtein. They serve in constructing covers of algebraic curves used in work by Oscar Zariski, Max Noether, and Alexander Grothendieck in moduli problems, and in explicit class field theory computations following approaches by Helmut Hasse and Takagi. Classical examples include additive analogues of cyclotomic extensions investigated by Emil Artin and explicit constructions used in research by Serge Lang.
Generalizations include Artin–Schreier–Witt theory, which extends the picture to Witt vectors and was developed in contexts involving Ernst Witt, connecting to Witt vectors and the frameworks of Serre and Grothendieck. The theory relates to Kummer theory for multiplicative extensions, to local class field theory as advanced by Tate and Iwasawa, and to the study of inseparable extensions explored by Zariski and Oscar Zariski’s contemporaries. Modern research links these ideas to étale cohomology, p-adic Hodge theory of Jean-Marc Fontaine, and to arithmetic geometry programs driven by Pierre Deligne and Alexander Grothendieck.
Category:Field theory