second supplementary law
E662762
The second supplementary law is a classical result in number theory that specifies the behavior of the Legendre symbol for the prime 2, complementing the quadratic reciprocity law.
Statements (29)
| Predicate | Object |
|---|---|
| instanceOf |
result in quadratic reciprocity
ⓘ
theorem in number theory ⓘ |
| appliesTo | odd prime p ⓘ |
| assumes | p is an odd prime ⓘ |
| characterizes | quadratic character of 2 modulo an odd prime ⓘ |
| complements | quadratic reciprocity law NERFINISHED ⓘ |
| concerns | quadratic residues modulo odd primes ⓘ |
| dependsOn |
definition of Legendre symbol
ⓘ
properties of quadratic residues modulo primes ⓘ |
| field | number theory ⓘ |
| hasAlternativeFormulation | (2/p) = (-1)^{(p^2-1)/8} for odd prime p ⓘ |
| hasAlternativeName | second supplementary law to quadratic reciprocity NERFINISHED ⓘ |
| holdsIn | ring of integers modulo p ⓘ |
| implies |
2 is a quadratic nonresidue modulo p if and only if p ≡ 3 or 5 (mod 8)
ⓘ
2 is a quadratic residue modulo p if and only if p ≡ 1 or 7 (mod 8) ⓘ |
| involves | prime number 2 ⓘ |
| isPartOf | supplementary laws to quadratic reciprocity ⓘ |
| isSupplementaryTo |
first supplementary law
ⓘ
third supplementary law ⓘ |
| isUsedIn |
computing Legendre symbols
ⓘ
elementary proofs in quadratic reciprocity ⓘ primality testing arguments ⓘ study of quadratic fields ⓘ |
| relatedTo |
Legendre symbol
NERFINISHED
ⓘ
quadratic reciprocity law NERFINISHED ⓘ |
| specifies | behavior of the Legendre symbol for the prime 2 ⓘ |
| states |
for an odd prime p, (2/p) = -1 if p ≡ ±3 (mod 8)
ⓘ
for an odd prime p, (2/p) = 1 if p ≡ ±1 (mod 8) ⓘ |
| uses | Legendre symbol (2/p) NERFINISHED ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.