cubic reciprocity law

E662763

reciprocity law result in algebraic number theory theorem in number theory

The cubic reciprocity law is a number-theoretic result that extends the quadratic reciprocity law to describe how prime numbers behave with respect to cubic residues in certain algebraic number fields.

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Statements (44)

Predicate	Object
instanceOf	reciprocity law ⓘ result in algebraic number theory ⓘ theorem in number theory ⓘ
appliesIn	certain algebraic number fields ⓘ cyclotomic fields of third roots of unity ⓘ
classification	higher power reciprocity law ⓘ
concerns	cubic residues ⓘ prime numbers ⓘ
describes	behavior of primes with respect to cubic residues ⓘ
expressedIn	ring Z[ω] where ω is a primitive cube root of unity ⓘ ring of Eisenstein integers ⓘ
extends	quadratic reciprocity law NERFINISHED ⓘ
field	number theory ⓘ
generalizes	quadratic reciprocity law NERFINISHED ⓘ
hasGeneralization	Artin reciprocity law NERFINISHED ⓘ Hilbert reciprocity law NERFINISHED ⓘ n-th power reciprocity laws ⓘ
historicalPrecursorOf	Artin reciprocity law NERFINISHED ⓘ general reciprocity law ⓘ
importance	fundamental in the theory of cyclotomic fields ⓘ key step toward general class field theory ⓘ
involves	prime ideals in Z[ω] ⓘ splitting of primes in cyclotomic extensions ⓘ
motivated	development of algebraic number theory ⓘ study of cyclotomic fields ⓘ
provedBy	Eisenstein NERFINISHED ⓘ Gauss NERFINISHED ⓘ Jacobi NERFINISHED ⓘ Kummer NERFINISHED ⓘ
relatedConcept	Eisenstein integers NERFINISHED ⓘ Legendre symbol NERFINISHED ⓘ cubic residue symbol ⓘ cyclotomic reciprocity ⓘ
relatedTo	Artin reciprocity NERFINISHED ⓘ class field theory NERFINISHED ⓘ higher reciprocity laws ⓘ quartic reciprocity law NERFINISHED ⓘ
studiedBy	David Hilbert NERFINISHED ⓘ Emil Artin NERFINISHED ⓘ
subfield	algebraic number theory ⓘ arithmetic of algebraic number fields ⓘ
typeOf	local-global principle for cubic residues ⓘ
uses	cubic residue symbol ⓘ
yearFirstProofApprox	19th century ⓘ

Referenced by (1)

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quadratic reciprocity law → generalizedBy → cubic reciprocity law ⓘ