Gauss sum

E662760

exponential sum mathematical concept

A Gauss sum is a finite exponential sum, often involving characters such as the Legendre symbol, that plays a central role in number theory and the study of quadratic and higher-order residues.

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Observed surface forms (1)

Surface form	Occurrences
Gauss sums	1

Statements (49)

Predicate	Object
instanceOf	exponential sum ⓘ mathematical concept ⓘ
appearsIn	Gauss’s Disquisitiones Arithmeticae NERFINISHED ⓘ
context	finite fields F_q ⓘ integers modulo n ⓘ
definitionInvolves	Dirichlet character NERFINISHED ⓘ Legendre symbol NERFINISHED ⓘ additive character ⓘ complex exponential ⓘ finite sum ⓘ multiplicative character ⓘ roots of unity ⓘ
field	number theory ⓘ
hasMagnitudeProperty	absolute value often equals square root of modulus ⓘ
hasSpecialCase	Jacobi sum NERFINISHED ⓘ Kloosterman sum NERFINISHED ⓘ cubic Gauss sum ⓘ quadratic Gauss sum ⓘ
namedAfter	Carl Friedrich Gauss NERFINISHED ⓘ
property	complex-valued ⓘ depends on character ⓘ depends on modulus ⓘ finite length ⓘ
relatedTo	Gauss reciprocity law NERFINISHED ⓘ Gauss sums over finite fields ⓘ Gaussian periods ⓘ Ramanujan sum NERFINISHED ⓘ Weil bound NERFINISHED ⓘ quadratic reciprocity ⓘ
specialCaseOf	character sum ⓘ exponential character sum ⓘ
usedIn	L-function theory ⓘ Weil conjectures NERFINISHED ⓘ algebraic number theory ⓘ analytic number theory ⓘ class field theory ⓘ coding theory ⓘ cryptography ⓘ finite field theory ⓘ higher-order residue theory ⓘ local class field theory ⓘ quadratic residue theory ⓘ representation theory of finite groups ⓘ
usedToCompute	Fourier transform of characters on finite abelian groups ⓘ exponential sums over finite fields ⓘ number of solutions of polynomial congruences ⓘ
usedToProve	non-vanishing of L-functions at certain points ⓘ properties of Dirichlet L-functions ⓘ quadratic reciprocity law NERFINISHED ⓘ

Referenced by (2)

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

quadratic reciprocity law → proofMethodsInclude → Gauss sum ⓘ

this entity surface form: Gauss sums

Legendre symbol → relatedConcept → Gauss sum ⓘ