idèle class group
E860117
The idèle class group is a fundamental arithmetic object in number theory that encodes global information about a number field via its idèles and plays a central role in class field theory.
All labels observed (1)
| Label | Occurrences |
|---|---|
| idèle class group canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T10389132 — 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: idèle class group Context triple: [Weil group, relatedTo, idèle class group]
-
A.
Hilbert class field
The Hilbert class field of a number field is its maximal unramified abelian extension, central in class field theory as it corresponds to the field’s ideal class group.
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B.
Dedekind ideal
A Dedekind ideal is a type of ideal in ring theory central to algebraic number theory, particularly in the study of Dedekind domains and unique factorization of ideals.
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C.
Weil group
The Weil group is an extension of the absolute Galois group introduced by André Weil to refine class field theory and play a central role in the formulation of the local and global Langlands correspondences.
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D.
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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E.
Brauer group
The Brauer group is an algebraic structure that classifies equivalence classes of central simple algebras over a field (or more general schemes), playing a key role in number theory, algebraic geometry, and cohomology.
- 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: idèle class group Target entity description: The idèle class group is a fundamental arithmetic object in number theory that encodes global information about a number field via its idèles and plays a central role in class field theory.
-
A.
Hilbert class field
The Hilbert class field of a number field is its maximal unramified abelian extension, central in class field theory as it corresponds to the field’s ideal class group.
-
B.
Dedekind ideal
A Dedekind ideal is a type of ideal in ring theory central to algebraic number theory, particularly in the study of Dedekind domains and unique factorization of ideals.
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C.
Weil group
The Weil group is an extension of the absolute Galois group introduced by André Weil to refine class field theory and play a central role in the formulation of the local and global Langlands correspondences.
-
D.
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.
-
E.
Brauer group
The Brauer group is an algebraic structure that classifies equivalence classes of central simple algebras over a field (or more general schemes), playing a key role in number theory, algebraic geometry, and cohomology.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
abelian group
ⓘ
locally compact group ⓘ mathematical object ⓘ topological group ⓘ |
| arisesFrom | restricted direct product of local multiplicative groups ⓘ |
| category | global arithmetic invariant ⓘ |
| constructedFrom |
idèle group
ⓘ
multiplicative group of a number field ⓘ |
| containsInformationAbout |
ideal class group
ⓘ
narrow class group ⓘ ray class groups ⓘ |
| definedAs | quotient of the idèle group by the multiplicative group of the field ⓘ |
| dependsOn | choice of number field K ⓘ |
| dualObject |
Größencharacters
ⓘ
Hecke characters NERFINISHED ⓘ |
| encodes | global arithmetic information of a number field ⓘ |
| field | number field ⓘ |
| generalizes | ideal class group ⓘ |
| hasComponent |
archimedean idèles
ⓘ
finite idèles ⓘ |
| hasSubgroup | connected component of identity at archimedean places ⓘ |
| introducedIn | class field theory ⓘ |
| localComponent | multiplicative group of a local field ⓘ |
| mapsTo | Galois group of maximal abelian extension via Artin map ⓘ |
| notation |
A_K^×/K^×
ⓘ
C_K ⓘ |
| property |
Hausdorff
NERFINISHED
ⓘ
locally compact abelian ⓘ σ-compact ⓘ |
| quotientBySubgroup | ideal class group ⓘ |
| relatedConcept |
Hilbert class field
NERFINISHED
ⓘ
adèle ⓘ idele group NERFINISHED ⓘ idèle ⓘ ray class field ⓘ |
| relatedTo | ideal class group ⓘ |
| roleInClassFieldTheory | Galois group of maximal abelian extension is isomorphic to a quotient of the idèle class group ⓘ |
| studiedBy | Claude Chevalley NERFINISHED ⓘ |
| symbolForField | C_K = A_K^×/K^× ⓘ |
| topology | quotient topology from the idèle group ⓘ |
| usedIn |
Artin reciprocity law
NERFINISHED
ⓘ
Tate’s thesis NERFINISHED ⓘ abelian extensions of number fields ⓘ definition of global L-functions via Hecke characters ⓘ global class field theory ⓘ harmonic analysis on adèle groups ⓘ |
| usedToClassify | finite abelian extensions of a number 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: idèle class group Description of subject: The idèle class group is a fundamental arithmetic object in number theory that encodes global information about a number field via its idèles and plays a central role in class field theory.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.