Jacobi sums
E734841
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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Jacobi sums canonical | 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.
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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.
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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.
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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.
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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
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 ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Jacobi sums Description of subject: 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.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.