Disambiguation evidence for Gauss’s lemma in number theory via surface form
"Gauss’s lemma (number theory)"
As subject (44)
Triples where this entity appears as subject under the
label "Gauss’s lemma (number theory)".
| Predicate | Object |
|---|---|
| appearsIn |
Disquisitiones Arithmeticae
ⓘ
surface form:
Gauss’s Disquisitiones Arithmeticae
|
| appliesTo | odd prime modulus ⓘ |
| assumes | a is an integer coprime to p ⓘ |
| assumes | p is an odd prime ⓘ |
| category | results about quadratic residues ⓘ |
| category | theorems about primes ⓘ |
| characterizes | quadratic non-residues via parity of negative multiples ⓘ |
| characterizes | quadratic residues via parity of negative multiples ⓘ |
| concerns | quadratic non-residues modulo an odd prime ⓘ |
| concerns | quadratic residues modulo an odd prime ⓘ |
| domain | modular arithmetic ⓘ |
| equates | Legendre symbol (a|p) with (−1)^n ⓘ |
| equivalentTo | certain combinatorial formulations of the Legendre symbol ⓘ |
| excludes | case where p divides a ⓘ |
| field | number theory ⓘ |
| gives | criterion for quadratic residuosity modulo an odd prime ⓘ |
| hasNotation | (a|p) for the Legendre symbol in its statement ⓘ |
| historicalPeriod | early 19th century ⓘ |
| holdsFor | integers a with gcd(a,p)=1 ⓘ |
| implies |
Gauss’s lemma in number theory
self-linksurface differs
ⓘ
surface form:
Legendre symbol takes values ±1 for a coprime to p
|
| instanceOf | lemma in number theory ⓘ |
| instanceOf | result in elementary number theory ⓘ |
| involves | counting negative representatives among reduced multiples ⓘ |
| involves | sequence of multiples a,2a,…,((p−1)/2)a modulo p ⓘ |
| namedAfter | Carl Friedrich Gauss ⓘ |
| oftenFormulatedWith | interval (−p/2,p/2] for representatives modulo p ⓘ |
| provides | practical method to compute Legendre symbols ⓘ |
| relatedTo |
Gauss’s lemma in number theory
self-linksurface differs
ⓘ
surface form:
Euler’s criterion
|
| relatedTo | quadratic reciprocity law ⓘ |
| relates | Legendre symbol ⓘ |
| relates | number of sign changes in a sequence of multiples ⓘ |
| requires | reduction of integers modulo p into a symmetric interval around 0 ⓘ |
| statement | Let p be an odd prime and a an integer coprime to p. Consider the numbers a,2a,3a,…,((p−1)/2)a reduced modulo p into the interval (−p/2,p/2]. Let n be the number of these residues that are negative. Then (a|p) = (−1)^n, where (a|p) is the Legendre symbol. ⓘ |
| subfield | elementary number theory ⓘ |
| subfield | quadratic number theory ⓘ |
| toolIn | computational number theory ⓘ |
| topic | Legendre symbol ⓘ |
| topic | quadratic residues ⓘ |
| typeOf | criterion for quadratic residuosity ⓘ |
| usedBy | elementary number theory textbooks ⓘ |
| usedFor | computing Legendre symbols ⓘ |
| usedFor | testing whether an integer is a quadratic residue modulo an odd prime ⓘ |
| usedIn | elementary proofs of properties of Legendre symbols ⓘ |
| usedIn | proofs of quadratic reciprocity ⓘ |