Artin reciprocity law
E537778
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.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Artin reciprocity law canonical | 3 |
| Artin reciprocity | 2 |
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.
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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.
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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.
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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 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 ⓘ |
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: Artin reciprocity law Description of subject: 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.
Referenced by (5)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Artin reciprocity
this entity surface form:
Artin reciprocity