global class field theory
E459561
Global class field theory is a branch of algebraic number theory that classifies finite abelian extensions of global fields (such as number fields) in terms of their arithmetic data, particularly via idele class groups and reciprocity maps.
All labels observed (3)
| Label | Occurrences |
|---|---|
| global class field theory canonical | 3 |
| Class Field Theory | 1 |
| class field theory | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T4597205 — 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: global class field theory Context triple: [Kronecker–Weber theorem, isPartOf, global class field theory]
-
A.
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.
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B.
Levine-Fricke Field
Levine-Fricke Field is the home softball stadium of the University of California, Berkeley Golden Bears, located on the university’s campus in Berkeley, California.
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C.
Grothendieck topology
A Grothendieck topology is an abstract framework in category theory that generalizes the notion of open covers in topology to define sheaves on arbitrary categories.
-
D.
Adeles and Algebraic Groups
"Adeles and Algebraic Groups" is a foundational mathematical work by André Weil that develops the theory of adeles and its deep connections with algebraic groups and number theory.
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E.
Grothendieck duality
Grothendieck duality is a foundational theory in algebraic geometry that generalizes classical Serre duality to a broad categorical and sheaf-theoretic framework for studying duality on schemes and morphisms.
- 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: global class field theory Target entity description: Global class field theory is a branch of algebraic number theory that classifies finite abelian extensions of global fields (such as number fields) in terms of their arithmetic data, particularly via idele class groups and reciprocity maps.
-
A.
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.
-
B.
Levine-Fricke Field
Levine-Fricke Field is the home softball stadium of the University of California, Berkeley Golden Bears, located on the university’s campus in Berkeley, California.
-
C.
Grothendieck topology
A Grothendieck topology is an abstract framework in category theory that generalizes the notion of open covers in topology to define sheaves on arbitrary categories.
-
D.
Adeles and Algebraic Groups
"Adeles and Algebraic Groups" is a foundational mathematical work by André Weil that develops the theory of adeles and its deep connections with algebraic groups and number theory.
-
E.
Grothendieck duality
Grothendieck duality is a foundational theory in algebraic geometry that generalizes classical Serre duality to a broad categorical and sheaf-theoretic framework for studying duality on schemes and morphisms.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
branch of algebraic number theory
ⓘ
mathematical theory ⓘ |
| appliesTo |
global function fields
ⓘ
number fields ⓘ |
| centralTheorem |
Artin reciprocity law
NERFINISHED
ⓘ
existence theorem for class fields ⓘ |
| characterizes | maximal abelian extension of a global field ⓘ |
| describes |
Galois group of maximal abelian extension as quotient of idele class group
ⓘ
abelian extensions via open subgroups of finite index in idele class group ⓘ |
| developedBy |
Claude Chevalley
NERFINISHED
ⓘ
Emil Artin NERFINISHED ⓘ Helmut Hasse NERFINISHED ⓘ Teiji Takagi NERFINISHED ⓘ |
| fieldOfStudy | algebraic number theory ⓘ |
| frameworkFor |
Langlands program (abelian case)
NERFINISHED
ⓘ
explicit class field theory ⓘ |
| generalizes |
Hilbert class field theory
NERFINISHED
ⓘ
Kronecker–Weber theorem NERFINISHED ⓘ |
| hasLocalAnalogue | local class field theory ⓘ |
| historicalRoot | Kronecker Jugendtraum NERFINISHED ⓘ |
| implies |
Kronecker–Weber theorem for abelian extensions of the rationals
NERFINISHED
ⓘ
existence of Hilbert class field for any number field ⓘ |
| relatedConcept |
class formation
ⓘ
cohomology of Galois groups ⓘ global field ⓘ idele topology ⓘ narrow class group ⓘ ray class group ⓘ |
| relates |
abelian Galois groups of global fields to idele class groups
ⓘ
ideal class groups to abelian extensions ⓘ |
| studies |
abelian extensions of global function fields
ⓘ
abelian extensions of number fields ⓘ finite abelian extensions of global fields ⓘ |
| typicalReference |
Artin–Tate: Class Field Theory
NERFINISHED
ⓘ
Cassels–Fröhlich: Algebraic Number Theory NERFINISHED ⓘ Neukirch: Algebraic Number Theory NERFINISHED ⓘ |
| usesConcept |
Artin reciprocity map
NERFINISHED
ⓘ
Chebotarev density theorem NERFINISHED ⓘ Frobenius automorphism NERFINISHED ⓘ Hilbert class field NERFINISHED ⓘ class group of a number field ⓘ global reciprocity law NERFINISHED ⓘ idele class characters ⓘ idele class group NERFINISHED ⓘ idele class group modulo connected component ⓘ idele group ⓘ norm map ⓘ ray class field ⓘ |
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: global class field theory Description of subject: Global class field theory is a branch of algebraic number theory that classifies finite abelian extensions of global fields (such as number fields) in terms of their arithmetic data, particularly via idele class groups and reciprocity maps.
Referenced by (5)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
class field theory
this entity surface form:
Class Field Theory