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.

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All labels observed (5)

Label	Occurrences
Artin reciprocity law canonical	11
Artin reciprocity	8
Artin reciprocity map	1
global Artin map	1
global reciprocity law	1

How this entity was disambiguated

This entity first appeared as the object of triple T5658018 — 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: Artin reciprocity law
Context triple: [Emil Artin, notableWork, Artin reciprocity law]

A. Chebotarev density theorem
The Chebotarev density theorem is a fundamental result in algebraic number theory that generalizes the prime number theorem to describe how often primes in a number field have a given Frobenius conjugacy class in its Galois group.
B. Hilbert’s twelfth problem
Hilbert’s twelfth problem is one of David Hilbert’s famous list of 23 problems, asking for a general explicit class field theory that would generate all abelian extensions of a given number field using special values of analytic functions.
C. Kronecker–Weber theorem
The Kronecker–Weber theorem is a fundamental result in algebraic number theory stating that every finite abelian extension of the rational numbers is contained in a cyclotomic field generated by roots of unity.
D. Iwasawa theory
Iwasawa theory is a branch of number theory that studies the growth of arithmetic invariants in infinite towers of number fields, particularly using p-adic methods.
E. Gauss’s lemma in number theory
Gauss’s lemma in number theory is a result that relates the Legendre symbol to the number of sign changes in a certain sequence of multiples, providing a practical criterion for determining quadratic residues modulo an odd prime.
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: Artin reciprocity law
Target entity description: 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.

A. Chebotarev density theorem
The Chebotarev density theorem is a fundamental result in algebraic number theory that generalizes the prime number theorem to describe how often primes in a number field have a given Frobenius conjugacy class in its Galois group.
B. Hilbert’s twelfth problem
Hilbert’s twelfth problem is one of David Hilbert’s famous list of 23 problems, asking for a general explicit class field theory that would generate all abelian extensions of a given number field using special values of analytic functions.
C. Kronecker–Weber theorem
The Kronecker–Weber theorem is a fundamental result in algebraic number theory stating that every finite abelian extension of the rational numbers is contained in a cyclotomic field generated by roots of unity.
D. Iwasawa theory
Iwasawa theory is a branch of number theory that studies the growth of arithmetic invariants in infinite towers of number fields, particularly using p-adic methods.
E. Gauss’s lemma in number theory
Gauss’s lemma in number theory is a result that relates the Legendre symbol to the number of sign changes in a certain sequence of multiples, providing a practical criterion for determining quadratic residues modulo an odd prime.
F. None of above. chosen

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 ⓘ linked to: global class field theory
generalizes	quadratic reciprocity ⓘ
hasVersion	ideal-theoretic formulation ⓘ idèle-theoretic formulation ⓘ
historicalPeriod	20th century mathematics ⓘ
implies	Kronecker–Weber theorem over the rationals ⓘ linked to: Kronecker–Weber theorem quadratic reciprocity law ⓘ
influenced	Langlands program ⓘ
involves	Artin symbol ⓘ linked to: Frobenius element Hecke characters ⓘ L-functions ⓘ unramified primes ⓘ
isPartOf	global class field theory ⓘ
namedAfter	Emil Artin ⓘ
provedBy	Emil Artin ⓘ
relatedTo	Chebotarev density theorem ⓘ Hilbert class field ⓘ global class field theory ⓘ local class field theory ⓘ ray class group ⓘ
relates	idele class groups to Galois groups of abelian extensions ⓘ
subfield	class field theory ⓘ linked to: global class field theory
usesConcept	Frobenius automorphism ⓘ linked to: Frobenius element global Artin map ⓘ linked to: Artin reciprocity law idele ⓘ idele class character ⓘ idele class group ⓘ linked to: idèle class group idele group ⓘ linked to: idèle class group local Artin map ⓘ norm map ⓘ ray class field ⓘ

How these facts were elicited