Jacobi sums

E734841

algebraic number theory concept number-theoretic function

Jacobi sums are special algebraic number theory constructs formed from character sums over finite fields or residue classes, widely used in primality testing and the study of cyclotomic fields.

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Statements (48)

Predicate	Object
instanceOf	algebraic number theory concept ⓘ number-theoretic function ⓘ
appearsIn	Adleman–Pomerance–Rumely primality test NERFINISHED ⓘ Cohen–Lenstra heuristics context via cyclotomic units ⓘ Jacobi sum primality test NERFINISHED ⓘ
definedOver	finite fields ⓘ residue class rings modulo n ⓘ
field	algebraic number theory ⓘ analytic number theory ⓘ number theory ⓘ
generalizationOf	certain binomial coefficient congruences ⓘ
hasProperty	Galois conjugates given by automorphisms of cyclotomic fields ⓘ algebraic integer values ⓘ expressible via Gauss sums ⓘ lie in cyclotomic fields ⓘ multiplicative in characters ⓘ satisfy congruence relations modulo prime ideals ⓘ
introducedIn	19th century ⓘ
namedAfter	Carl Gustav Jacob Jacobi NERFINISHED ⓘ
relatedTo	Dirichlet characters NERFINISHED ⓘ Gauss sums NERFINISHED ⓘ Jacobi sums of order l NERFINISHED ⓘ Stickelberger elements NERFINISHED ⓘ cyclotomic characters ⓘ cyclotomic polynomials ⓘ multiplicative characters of finite fields ⓘ
satisfies	functional equations under complex conjugation ⓘ multiplicative relations with Gauss sums ⓘ orthogonality-type relations for characters ⓘ
typicalDomain	pairs of multiplicative characters ⓘ tuples of multiplicative characters ⓘ
usedBy	E. Artin NERFINISHED ⓘ H. Hasse NERFINISHED ⓘ S. Lang NERFINISHED ⓘ
usedFor	bounding character sums ⓘ computing ideal class groups in cyclotomic fields ⓘ computing local L-factors ⓘ constructing explicit units in abelian extensions of Q ⓘ studying exponential sums over finite fields ⓘ studying ramification in cyclotomic extensions ⓘ
usedIn	Gauss sum factorizations ⓘ L-functions NERFINISHED ⓘ Weil conjectures over finite fields NERFINISHED ⓘ construction of cyclotomic units ⓘ cyclotomic fields ⓘ primality testing ⓘ study of class groups of cyclotomic fields ⓘ
valueType	elements of cyclotomic number fields ⓘ

Referenced by (1)

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Adleman–Pomerance–Rumely primality test → uses → Jacobi sums ⓘ