Artin–Schreier theory

E537780

mathematical theory theory in algebraic number theory theory in field theory

Artin–Schreier theory is a branch of algebraic number theory and field theory that characterizes cyclic extensions of prime degree in fields of characteristic p using additive polynomials.

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Statements (46)

Predicate	Object
instanceOf	mathematical theory ⓘ theory in algebraic number theory ⓘ theory in field theory ⓘ
appliesTo	fields of characteristic p ⓘ
assumes	base field of characteristic p > 0 ⓘ
characterizes	cyclic extensions of prime degree in characteristic p ⓘ
classifies	degree p extensions of fields of characteristic p up to isomorphism ⓘ
concerns	additive structure of fields of characteristic p ⓘ extensions with exponent p in the Galois group ⓘ
describes	cyclic Galois extensions with Galois group of order p ⓘ extensions obtained by adjoining roots of Artin–Schreier equations ⓘ
frameworkFor	describing wild ramification in characteristic p ⓘ
generalizes	Kummer theory to characteristic p ⓘ
hasAnalogue	Artin–Schreier–Witt theory NERFINISHED ⓘ
hasApplication	classification of p-extensions of local fields of characteristic p ⓘ coding theory via function fields over finite fields ⓘ construction of Artin–Schreier curves ⓘ explicit class field theory for global function fields ⓘ
involves	Artin–Schreier polynomials NERFINISHED ⓘ equations of the form X^p − X − a ⓘ
isAnalogousTo	Kummer theory for multiplicative extensions NERFINISHED ⓘ
isDevelopedIn	20th-century algebra ⓘ
isDiscussedIn	texts on Galois theory ⓘ texts on algebraic number theory ⓘ texts on field theory ⓘ
isPartOf	local class field theory in characteristic p ⓘ
isRelatedTo	Witt vectors via Artin–Schreier–Witt theory ⓘ finite fields and their extensions ⓘ ramification theory in characteristic p ⓘ étale covers in characteristic p ⓘ
isUsedIn	explicit description of Galois groups of p-extensions ⓘ the construction of covers of algebraic curves in characteristic p ⓘ the study of function fields over finite fields ⓘ
namedAfter	Emil Artin NERFINISHED ⓘ Otto Schreier NERFINISHED ⓘ
provides	a correspondence between degree p cyclic extensions and additive polynomial equations ⓘ explicit generators for cyclic p-extensions ⓘ
relatesTo	Galois theory NERFINISHED ⓘ additive Galois cohomology in characteristic p ⓘ additive characters of finite fields ⓘ cohomology group H^1(Gal(K^sep/K), ℤ/pℤ) ⓘ cyclic extensions ⓘ
studies	Galois extensions of degree p ⓘ
uses	additive polynomials ⓘ the Frobenius endomorphism NERFINISHED ⓘ the map x ↦ x^p − x ⓘ

Referenced by (2)

Full triples — surface form annotated when it differs from this entity's canonical label.

Emil Artin → notableWork → Artin–Schreier theory ⓘ

Hilbert’s seventeenth problem → solutionMethod → Artin–Schreier theory ⓘ