Artin reciprocity law

E537778

result in class field theory theorem in number theory

The Artin reciprocity law is a fundamental theorem in class field theory that generalizes quadratic reciprocity by describing abelian extensions of number fields in terms of characters of their idele class groups.

Jump to: Surface forms Statements Referenced by

Observed surface forms (1)

Surface form	Occurrences
Artin reciprocity	2

Statements (45)

Predicate	Object
instanceOf	result in class field theory ⓘ theorem in number theory ⓘ
asserts	kernel of the Artin map corresponds to norms from the abelian extension ⓘ
assertsExistenceOf	canonical surjective homomorphism from the idele class group to the abelianized Galois group ⓘ
characterizationOf	finite abelian extensions of a number field ⓘ
characterizes	maximal abelian extension of a number field via its idele class group ⓘ
codomain	Galois group of a finite abelian extension ⓘ
concerns	abelianized absolute Galois group of a number field ⓘ finite abelian extensions unramified outside a modulus ⓘ
describes	abelian extensions of number fields ⓘ
domain	idele class group of a number field ⓘ
field	algebraic number theory ⓘ number theory ⓘ
framework	Galois theory ⓘ class field theory NERFINISHED ⓘ
generalizes	quadratic reciprocity ⓘ
hasVersion	ideal-theoretic formulation ⓘ idèle-theoretic formulation ⓘ
historicalPeriod	20th century mathematics ⓘ
implies	Kronecker–Weber theorem over the rationals NERFINISHED ⓘ quadratic reciprocity law ⓘ
influenced	Langlands program NERFINISHED ⓘ
involves	Artin symbol NERFINISHED ⓘ Hecke characters NERFINISHED ⓘ L-functions NERFINISHED ⓘ unramified primes ⓘ
isPartOf	global class field theory NERFINISHED ⓘ
namedAfter	Emil Artin NERFINISHED ⓘ
provedBy	Emil Artin NERFINISHED ⓘ
relatedTo	Chebotarev density theorem NERFINISHED ⓘ Hilbert class field NERFINISHED ⓘ global class field theory NERFINISHED ⓘ local class field theory ⓘ ray class group ⓘ
relates	idele class groups to Galois groups of abelian extensions ⓘ
subfield	class field theory NERFINISHED ⓘ
usesConcept	Frobenius automorphism NERFINISHED ⓘ global Artin map NERFINISHED ⓘ idele ⓘ idele class character ⓘ idele class group NERFINISHED ⓘ idele group NERFINISHED ⓘ local Artin map ⓘ norm map ⓘ ray class field ⓘ

Referenced by (5)

Full triples — surface form annotated when it differs from this entity's canonical label.

quadratic reciprocity law → generalizedBy → Artin reciprocity law ⓘ

Emil Artin → knownFor → Artin reciprocity law ⓘ

this entity surface form: Artin reciprocity

Emil Artin → notableWork → Artin reciprocity law ⓘ

Weil group → relatedConcept → Artin reciprocity law ⓘ

this entity surface form: Artin reciprocity

Hasse norm theorem → relatedTo → Artin reciprocity law ⓘ