quartic reciprocity law
E662764
The quartic reciprocity law is a number-theoretic result that extends quadratic reciprocity by characterizing when an integer is a fourth power residue modulo an odd prime, using properties of Gaussian integers and higher power residue symbols.
All labels observed (1)
| Label | Occurrences |
|---|---|
| quartic reciprocity law canonical | 2 |
How this entity was disambiguated
This entity first appeared as the object of triple T7420218 — 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: quartic reciprocity law Context triple: [quadratic reciprocity law, generalizedBy, quartic reciprocity law]
-
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.
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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C.
Jacobi’s four-square theorem
Jacobi’s four-square theorem is a fundamental result in number theory that gives a precise formula for the number of ways an integer can be expressed as a sum of four squares.
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D.
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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E.
Lagrange's four-square theorem
Lagrange's four-square theorem is a fundamental result in number theory stating that every natural number can be expressed as the sum of four integer squares.
- 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: quartic reciprocity law Target entity description: The quartic reciprocity law is a number-theoretic result that extends quadratic reciprocity by characterizing when an integer is a fourth power residue modulo an odd prime, using properties of Gaussian integers and higher power residue symbols.
-
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.
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.
-
C.
Jacobi’s four-square theorem
Jacobi’s four-square theorem is a fundamental result in number theory that gives a precise formula for the number of ways an integer can be expressed as a sum of four squares.
-
D.
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.
-
E.
Lagrange's four-square theorem
Lagrange's four-square theorem is a fundamental result in number theory stating that every natural number can be expressed as the sum of four integer squares.
- F. None of above. chosen
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
number-theoretic reciprocity law
ⓘ
theorem in number theory ⓘ |
| characterizes | when an integer is a fourth power residue modulo an odd prime ⓘ |
| classification | higher power reciprocity law ⓘ |
| concerns |
fourth power residues modulo primes
ⓘ
primes congruent to 1 modulo 4 ⓘ representation of primes as sums of two squares ⓘ |
| expressedUsing |
congruence conditions on primes in Z[i]
ⓘ
quartic residue symbol (a|p)_4 ⓘ |
| extends | quadratic reciprocity law ⓘ |
| field | number theory ⓘ |
| generalizedBy |
Artin reciprocity law
ⓘ
class field theory ⓘ
linked to:
global class field theory
|
| historicalPrecursorOf | class field theory ⓘ |
| involves |
Dirichlet characters of order 4
ⓘ
Gauss sums ⓘ
linked to:
Gauss sum
Legendre symbol generalizations ⓘ congruences modulo powers of primes ⓘ norms in quadratic imaginary fields ⓘ primary primes in Z[i] ⓘ prime ideals in the ring of Gaussian integers ⓘ |
| motivationFor | development of higher reciprocity laws ⓘ |
| proofTechnique |
Gauss sums
ⓘ
linked to:
Gauss sum
algebraic number theory methods ⓘ ideal-theoretic methods ⓘ |
| provedBy | Carl Friedrich Gauss ⓘ |
| relatedTo |
Eisenstein reciprocity law
ⓘ
Hilbert symbol ⓘ biquadratic residues ⓘ cubic reciprocity law ⓘ cyclotomic fields ⓘ higher reciprocity laws ⓘ local reciprocity law ⓘ quartic Gauss sums ⓘ |
| requiresPrerequisite |
basic algebraic number theory
ⓘ
properties of Gaussian integers ⓘ quadratic reciprocity ⓘ |
| statedIn | ring of Gaussian integers Z[i] ⓘ |
| status | proven theorem ⓘ |
| subfield |
algebraic number theory
ⓘ
elementary number theory ⓘ |
| topicOf |
advanced textbooks on algebraic number theory
ⓘ
research in explicit class field theory ⓘ |
| uses |
Gaussian integers
ⓘ
higher power residue symbols ⓘ quartic residue symbol ⓘ |
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
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Input
Subject: quartic reciprocity law Description of subject: The quartic reciprocity law is a number-theoretic result that extends quadratic reciprocity by characterizing when an integer is a fourth power residue modulo an odd prime, using properties of Gaussian integers and higher power residue symbols.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.