cubic reciprocity law
E662763
The cubic reciprocity law is a number-theoretic result that extends the quadratic reciprocity law to describe how prime numbers behave with respect to cubic residues in certain algebraic number fields.
All labels observed (1)
| Label | Occurrences |
|---|---|
| cubic reciprocity law canonical | 2 |
How this entity was disambiguated
This entity first appeared as the object of triple T7420217 — 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: cubic reciprocity law Context triple: [quadratic reciprocity law, generalizedBy, cubic reciprocity law]
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A.
quadratic reciprocity law
The quadratic reciprocity law is a fundamental theorem in number theory that characterizes when a quadratic equation modulo one odd prime has solutions in terms of solvability modulo another, revealing a deep symmetry between primes.
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B.
Artin reciprocity law
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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C.
cyclotomic fields
Cyclotomic fields are number fields obtained by adjoining complex roots of unity to the rationals, playing a central role in algebraic number theory and classical geometric constructibility.
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D.
Higher composition laws I–IV
Higher composition laws I–IV is a landmark four-part series of papers by Manjul Bhargava that generalizes Gauss’s composition of binary quadratic forms and develops new structures and methods in algebraic number theory.
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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: cubic reciprocity law Target entity description: The cubic reciprocity law is a number-theoretic result that extends the quadratic reciprocity law to describe how prime numbers behave with respect to cubic residues in certain algebraic number fields.
-
A.
quadratic reciprocity law
The quadratic reciprocity law is a fundamental theorem in number theory that characterizes when a quadratic equation modulo one odd prime has solutions in terms of solvability modulo another, revealing a deep symmetry between primes.
-
B.
Artin reciprocity law
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.
-
C.
cyclotomic fields
Cyclotomic fields are number fields obtained by adjoining complex roots of unity to the rationals, playing a central role in algebraic number theory and classical geometric constructibility.
-
D.
Higher composition laws I–IV
Higher composition laws I–IV is a landmark four-part series of papers by Manjul Bhargava that generalizes Gauss’s composition of binary quadratic forms and develops new structures and methods in algebraic number theory.
-
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 (44)
| Predicate | Object |
|---|---|
| instanceOf |
reciprocity law
ⓘ
result in algebraic number theory ⓘ theorem in number theory ⓘ |
| appliesIn |
certain algebraic number fields
ⓘ
cyclotomic fields of third roots of unity ⓘ |
| classification | higher power reciprocity law ⓘ |
| concerns |
cubic residues
ⓘ
prime numbers ⓘ |
| describes | behavior of primes with respect to cubic residues ⓘ |
| expressedIn |
ring Z[ω] where ω is a primitive cube root of unity
ⓘ
ring of Eisenstein integers ⓘ |
| extends | quadratic reciprocity law ⓘ |
| field | number theory ⓘ |
| generalizes | quadratic reciprocity law ⓘ |
| hasGeneralization |
Artin reciprocity law
ⓘ
Hilbert reciprocity law ⓘ
linked to:
Hilbert symbol
n-th power reciprocity laws ⓘ |
| historicalPrecursorOf |
Artin reciprocity law
ⓘ
general reciprocity law ⓘ |
| importance |
fundamental in the theory of cyclotomic fields
ⓘ
key step toward general class field theory ⓘ |
| involves |
prime ideals in Z[ω]
ⓘ
splitting of primes in cyclotomic extensions ⓘ |
| motivated |
development of algebraic number theory
ⓘ
study of cyclotomic fields ⓘ |
| provedBy |
Eisenstein
ⓘ
Gauss ⓘ
linked to:
Carl Friedrich Gauss
Jacobi ⓘ Kummer ⓘ |
| relatedConcept |
Eisenstein integers
ⓘ
Legendre symbol ⓘ cubic residue symbol ⓘ cyclotomic reciprocity ⓘ |
| relatedTo |
Artin reciprocity
ⓘ
linked to:
Artin reciprocity law
class field theory ⓘ
linked to:
global class field theory
higher reciprocity laws ⓘ quartic reciprocity law ⓘ |
| studiedBy |
David Hilbert
ⓘ
Emil Artin ⓘ |
| subfield |
algebraic number theory
ⓘ
arithmetic of algebraic number fields ⓘ |
| typeOf | local-global principle for cubic residues ⓘ |
| uses | cubic residue symbol ⓘ |
| yearFirstProofApprox | 19th century ⓘ |
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: cubic reciprocity law Description of subject: The cubic reciprocity law is a number-theoretic result that extends the quadratic reciprocity law to describe how prime numbers behave with respect to cubic residues in certain algebraic number fields.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.