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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All labels observed (3)

Label	Occurrences
Artin–Schreier theory canonical	2
Artin–Schreier polynomials	1
Artin–Schreier–Witt theory	1

How this entity was disambiguated

This entity first appeared as the object of triple T5658021 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.

NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07

Target entity: Artin–Schreier theory
Context triple: [Emil Artin, notableWork, Artin–Schreier theory]

A. Kummer theory
Kummer theory is a branch of algebraic number theory that studies abelian extensions of fields, especially cyclotomic and radical extensions, using properties of roots of unity and ideal class groups.
B. Algebraic Groups and Class Fields
"Algebraic Groups and Class Fields" is a influential mathematical monograph that develops the deep connections between algebraic group theory and class field theory within number theory and arithmetic geometry.
C. Galois theory
Galois theory is a branch of abstract algebra that studies field extensions and polynomial equations through the structure of their associated symmetry groups.
D. Hasse–Arf theorem
The Hasse–Arf theorem is a fundamental result in algebraic number theory that precisely characterizes the jumps in the ramification filtration of abelian extensions of local fields, showing they occur at integer values.
E. Hilbert’s irreducibility theorem
Hilbert’s irreducibility theorem is a fundamental result in number theory and algebraic geometry that ensures many polynomial equations with parameterized coefficients retain irreducibility for infinitely many specializations of those parameters.
F. None of above. chosen
G. Unsure - the case is ambiguous/there is not enough information to decide.

NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07

Target entity: Artin–Schreier theory
Target entity description: 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.

A. Kummer theory
Kummer theory is a branch of algebraic number theory that studies abelian extensions of fields, especially cyclotomic and radical extensions, using properties of roots of unity and ideal class groups.
B. Algebraic Groups and Class Fields
"Algebraic Groups and Class Fields" is a influential mathematical monograph that develops the deep connections between algebraic group theory and class field theory within number theory and arithmetic geometry.
C. Galois theory
Galois theory is a branch of abstract algebra that studies field extensions and polynomial equations through the structure of their associated symmetry groups.
D. Hasse–Arf theorem
The Hasse–Arf theorem is a fundamental result in algebraic number theory that precisely characterizes the jumps in the ramification filtration of abelian extensions of local fields, showing they occur at integer values.
E. Hilbert’s irreducibility theorem
Hilbert’s irreducibility theorem is a fundamental result in number theory and algebraic geometry that ensures many polynomial equations with parameterized coefficients retain irreducibility for infinitely many specializations of those parameters.
F. None of above. chosen

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 ⓘ linked to: Artin–Schreier theory
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 ⓘ linked to: Artin–Schreier theory equations of the form X^p − X − a ⓘ
isAnalogousTo	Kummer theory for multiplicative extensions ⓘ linked to: Kummer theory
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 ⓘ Otto Schreier ⓘ
provides	a correspondence between degree p cyclic extensions and additive polynomial equations ⓘ explicit generators for cyclic p-extensions ⓘ
relatesTo	Galois theory ⓘ 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 ⓘ linked to: Frobenius endomorphism the map x ↦ x^p − x ⓘ

How these facts were elicited

Referenced by (4)

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 ⓘ

Artin–Schreier theory → involves → Artin–Schreier polynomials ⓘ

linked to: Artin–Schreier theory

Artin–Schreier theory → hasAnalogue → Artin–Schreier–Witt theory ⓘ