Artin–Schreier theory
E537780
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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Artin–Schreier theory canonical | 2 |
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 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 ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Artin–Schreier theory Description of subject: 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.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.