quadratic reciprocity law
E171226
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.
All labels observed (4)
| Label | Occurrences |
|---|---|
| quadratic reciprocity law canonical | 7 |
| Gauss reciprocity law | 1 |
| Gauss's law of quadratic reciprocity | 1 |
| law of quadratic reciprocity (formulation) | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T1489725 — 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: quadratic reciprocity law Context triple: [Gauss’s lemma (number theory), relatedTo, quadratic reciprocity law]
-
A.
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.
-
B.
Fermat's theorem on sums of two squares
Fermat's theorem on sums of two squares is a result in number theory stating exactly which prime numbers (and, more generally, which integers) can be expressed as the sum of two perfect squares.
-
C.
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.
-
D.
Gaussian periods
Gaussian periods are special algebraic sums of roots of unity that play a key role in number theory, particularly in constructing regular polygons like the 17-gon with straightedge and compass and in understanding cyclotomic fields.
-
E.
Disquisitiones Arithmeticae
Disquisitiones Arithmeticae is a foundational 1801 treatise on number theory that systematically developed the subject and introduced many of its central concepts and methods.
- 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: quadratic reciprocity law Target entity description: 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.
-
A.
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.
-
B.
Fermat's theorem on sums of two squares
Fermat's theorem on sums of two squares is a result in number theory stating exactly which prime numbers (and, more generally, which integers) can be expressed as the sum of two perfect squares.
-
C.
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.
-
D.
Gaussian periods
Gaussian periods are special algebraic sums of roots of unity that play a key role in number theory, particularly in constructing regular polygons like the 17-gon with straightedge and compass and in understanding cyclotomic fields.
-
E.
Disquisitiones Arithmeticae
Disquisitiones Arithmeticae is a foundational 1801 treatise on number theory that systematically developed the subject and introduced many of its central concepts and methods.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
reciprocity law
ⓘ
theorem in number theory ⓘ |
| appliesTo | odd primes ⓘ |
| centralConcept | quadratic character modulo a prime ⓘ |
| characterizes | solvability of quadratic congruences modulo primes ⓘ |
| concerns |
Legendre symbol
ⓘ
prime numbers ⓘ quadratic residues ⓘ |
| doesNotDirectlyApplyTo | p = 2 or q = 2 ⓘ |
| expresses | symmetry between primes in quadratic residue behavior ⓘ |
| extendedBy | supplementary laws for p = 2 ⓘ |
| field | number theory ⓘ |
| firstCompleteProofBy | Carl Friedrich Gauss ⓘ |
| firstSupplementStates | for odd prime p, (-1/p) = (-1)^{(p-1)/2} ⓘ |
| formalizedUsing |
Dirichlet characters
ⓘ
Kronecker symbol ⓘ |
| generalizedBy |
Artin reciprocity law
ⓘ
cubic reciprocity law ⓘ higher reciprocity laws ⓘ quartic reciprocity law ⓘ |
| hasManyProofs | yes ⓘ |
| hasSupplement |
first supplementary law
ⓘ
second supplementary law ⓘ |
| historicalName |
Gauss’s golden theorem
ⓘ
linked to:
Gauss’s lemma in number theory
|
| holdsFor | distinct odd primes p and q ⓘ |
| implies | criteria for quadratic residues modulo primes ⓘ |
| importance | fundamental theorem of elementary number theory ⓘ |
| influencedDevelopmentOf |
Galois theory
ⓘ
algebraic number theory ⓘ |
| introducedBy | Leonhard Euler ⓘ |
| involves | parity of (p-1)/2 and (q-1)/2 ⓘ |
| numberOfProofsByGauss | at least 8 ⓘ |
| proofMethodsInclude |
Galois theory
ⓘ
Gauss sums ⓘ
linked to:
Gauss sum
class field theory ⓘ
linked to:
global class field theory
genus theory ⓘ lattice point counting ⓘ |
| publishedIn | Disquisitiones Arithmeticae ⓘ |
| relatedTo |
Hilbert symbol
ⓘ
linked to:
Hasse invariant
class field theory ⓘ local-global principles ⓘ |
| relates | (p/q) and (q/p) Legendre symbols ⓘ |
| secondSupplementStates | for odd prime p, (2/p) = (-1)^{(p^2-1)/8} ⓘ |
| states | for distinct odd primes p and q, (p/q)(q/p) = (-1)^((p-1)(q-1)/4) ⓘ |
| usedFor |
computing Legendre symbols efficiently
ⓘ
determining solvability of x^2 ≡ a (mod p) ⓘ |
| yearFirstCompleteProof | 1801 ⓘ |
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: quadratic reciprocity law Description of subject: 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.
Referenced by (10)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject linked to:
Gauss’s lemma in number theory
linked to: quadratic reciprocity law
subject linked to:
Legendre
linked to: quadratic reciprocity law