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.
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 NERFINISHED ⓘ |
| 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 NERFINISHED ⓘ |
| field | number theory ⓘ |
| generalizedBy |
Artin reciprocity law
NERFINISHED
ⓘ
class field theory NERFINISHED ⓘ |
| historicalPrecursorOf | class field theory ⓘ |
| involves |
Dirichlet characters of order 4
ⓘ
Gauss sums NERFINISHED ⓘ 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
NERFINISHED
ⓘ
algebraic number theory methods ⓘ ideal-theoretic methods ⓘ |
| provedBy | Carl Friedrich Gauss NERFINISHED ⓘ |
| relatedTo |
Eisenstein reciprocity law
NERFINISHED
ⓘ
Hilbert symbol NERFINISHED ⓘ biquadratic residues ⓘ cubic reciprocity law NERFINISHED ⓘ 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 ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.