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.
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 ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.