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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All labels observed (3)

Label	Occurrences
Jacobi sum	1
Jacobi sums canonical	1
Jacobi sums of order l	1

How this entity was disambiguated

This entity first appeared as the object of triple T8448911 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.

NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07

Target entity: Jacobi sums
Context triple: [Adleman–Pomerance–Rumely primality test, uses, Jacobi sums]

A. Gaussian periods
Gaussian periods are special algebraic sums of roots of unity that play a key role in number theory, particularly in constructing regular polygons like the 17-gon with straightedge and compass and in understanding cyclotomic fields.
B. Ramanujan’s sum
Ramanujan’s sum is a number-theoretic function introduced by Srinivasa Ramanujan, expressing certain periodic arithmetic functions as finite trigonometric sums over primitive roots of unity.
C. Jacobi’s four-square theorem
Jacobi’s four-square theorem is a fundamental result in number theory that gives a precise formula for the number of ways an integer can be expressed as a sum of four squares.
D. Jacobi symbol
The Jacobi symbol is a number-theoretic function that generalizes the Legendre symbol and plays a key role in quadratic residues and primality testing in modular arithmetic.
E. Kummer congruences
Kummer congruences are number-theoretic relations describing how special values of Bernoulli numbers and related arithmetic functions behave modulo powers of primes, foundational in the study of p-adic L-functions and cyclotomic fields.
F. None of above. chosen
G. Unsure - the case is ambiguous/there is not enough information to decide.

NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07

Target entity: Jacobi sums
Target entity description: 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.

A. Gaussian periods
Gaussian periods are special algebraic sums of roots of unity that play a key role in number theory, particularly in constructing regular polygons like the 17-gon with straightedge and compass and in understanding cyclotomic fields.
B. Ramanujan’s sum
Ramanujan’s sum is a number-theoretic function introduced by Srinivasa Ramanujan, expressing certain periodic arithmetic functions as finite trigonometric sums over primitive roots of unity.
C. Jacobi’s four-square theorem
Jacobi’s four-square theorem is a fundamental result in number theory that gives a precise formula for the number of ways an integer can be expressed as a sum of four squares.
D. Jacobi symbol
The Jacobi symbol is a number-theoretic function that generalizes the Legendre symbol and plays a key role in quadratic residues and primality testing in modular arithmetic.
E. Kummer congruences
Kummer congruences are number-theoretic relations describing how special values of Bernoulli numbers and related arithmetic functions behave modulo powers of primes, foundational in the study of p-adic L-functions and cyclotomic fields.
F. None of above. chosen

Statements (48)

Predicate	Object
instanceOf	algebraic number theory concept ⓘ number-theoretic function ⓘ
appearsIn	Adleman–Pomerance–Rumely primality test ⓘ Cohen–Lenstra heuristics context via cyclotomic units ⓘ Jacobi sum primality test ⓘ
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 ⓘ
relatedTo	Dirichlet characters ⓘ Gauss sums ⓘ linked to: Gauss sum Jacobi sums of order l ⓘ linked to: Jacobi sums Stickelberger elements ⓘ 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 ⓘ linked to: Emil Artin H. Hasse ⓘ linked to: Helmut Hasse S. Lang ⓘ linked to: Serge Lang
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 ⓘ Weil conjectures over finite fields ⓘ linked to: Weil conjectures construction of cyclotomic units ⓘ cyclotomic fields ⓘ primality testing ⓘ study of class groups of cyclotomic fields ⓘ
valueType	elements of cyclotomic number fields ⓘ

How these facts were elicited

Referenced by (3)

Full triples — surface form annotated when it differs from this entity's canonical label.

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

Gauss sum → hasSpecialCase → Jacobi sum ⓘ

linked to: Jacobi sums

Jacobi sums → relatedTo → Jacobi sums of order l ⓘ