local class field theory
E753158
Local class field theory is a branch of number theory that describes the abelian extensions of local fields (such as p-adic fields) in terms of their multiplicative groups via reciprocity maps.
All labels observed (1)
| Label | Occurrences |
|---|---|
| local class field theory canonical | 5 |
How this entity was disambiguated
This entity first appeared as the object of triple T8733552 — 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: local class field theory Context triple: [Hasse–Arf theorem, context, local class field theory]
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A.
global class field theory
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.
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B.
Local Fields
"Local Fields" is a foundational mathematical text by Jean-Pierre Serre that develops the theory of local fields and their applications in number theory and algebraic geometry.
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C.
Klass
Klass is a surname most notably associated with Philip J. Klass, an American journalist and prominent UFO skeptic.
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D.
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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E.
Furtwängler’s theorem in class field theory
Furtwängler’s theorem in class field theory is a fundamental result in algebraic number theory that refines the principal ideal theorem by describing how ideal classes capitulate (become principal) in certain abelian extensions of number fields.
- 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: local class field theory Target entity description: Local class field theory is a branch of number theory that describes the abelian extensions of local fields (such as p-adic fields) in terms of their multiplicative groups via reciprocity maps.
-
A.
global class field theory
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.
-
B.
Local Fields
"Local Fields" is a foundational mathematical text by Jean-Pierre Serre that develops the theory of local fields and their applications in number theory and algebraic geometry.
-
C.
Klass
Klass is a surname most notably associated with Philip J. Klass, an American journalist and prominent UFO skeptic.
-
D.
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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E.
Furtwängler’s theorem in class field theory
Furtwängler’s theorem in class field theory is a fundamental result in algebraic number theory that refines the principal ideal theorem by describing how ideal classes capitulate (become principal) in certain abelian extensions of number fields.
- F. None of above. chosen
Statements (50)
| Predicate | Object |
|---|---|
| instanceOf |
branch of number theory
ⓘ
class field theory ⓘ mathematical theory ⓘ |
| appliesTo |
finite extensions of F_p((T))
ⓘ
finite extensions of Q_p ⓘ non-archimedean local fields ⓘ |
| characterizedBy |
compatibility with norm maps and restriction maps
ⓘ
description of higher ramification groups via unit filtration ⓘ existence theorem for abelian extensions of local fields ⓘ functorial correspondence between open subgroups of multiplicative group and finite abelian extensions ⓘ isomorphism between profinite completion of multiplicative group and abelianized Galois group ⓘ |
| describes | abelian extensions of local fields in terms of multiplicative groups ⓘ |
| developedBy |
Claude Chevalley
ⓘ
Emil Artin ⓘ Helmut Hasse ⓘ Shokichi Iyanaga ⓘ Teiji Takagi NERFINISHED ⓘ |
| fieldOfStudy | number theory ⓘ |
| formalizedUsing |
Galois cohomology
ⓘ
profinite groups ⓘ topological groups ⓘ |
| generalizes | properties of cyclotomic extensions of Q_p ⓘ |
| hasApplication |
classification of finite abelian extensions of local fields
ⓘ
computation of local Artin L-factors ⓘ study of ramification in local Galois extensions ⓘ |
| hasTheorem |
Hasse–Arf theorem (in the abelian case)
ⓘ
linked to:
Hasse–Arf theorem
existence theorem of local class field theory ⓘ local reciprocity law ⓘ |
| influenced |
local Langlands program
ⓘ
linked to:
Langlands program
modern algebraic number theory ⓘ |
| relatedTo |
Artin reciprocity
ⓘ
linked to:
Artin reciprocity law
Kronecker–Weber theorem ⓘ Lubin–Tate theory ⓘ
linked to:
Lubin–Tate formal groups
global class field theory ⓘ local Langlands correspondence ⓘ |
| studies | abelian extensions of local fields ⓘ |
| usesConcept |
Brauer group
ⓘ
Galois group of an abelian extension ⓘ Hasse invariant ⓘ finite extensions of F_p((T)) ⓘ finite extensions of Q_p ⓘ idèle group of a local field ⓘ inertia group ⓘ local fields ⓘ local reciprocity map ⓘ multiplicative group of a local field ⓘ norm map ⓘ p-adic fields ⓘ ramification group ⓘ reciprocity map ⓘ |
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: local class field theory Description of subject: Local class field theory is a branch of number theory that describes the abelian extensions of local fields (such as p-adic fields) in terms of their multiplicative groups via reciprocity maps.
Referenced by (5)
Full triples — surface form annotated when it differs from this entity's canonical label.