Gauss sum

Let p be a prime. Let χ be any multiplicative groupMathworldPlanetmath characterPlanetmathPlanetmathPlanetmath on /p (that is, any group homomorphismMathworldPlanetmath of multiplicative groups (/p)××). For any a/p, the complex numberMathworldPlanetmathPlanetmath


is called a Gauss sum on /p associated to χ.

In general, the equation ga(χ)=χ(a-1)g1(χ) (for nontrivial a and χ) reduces the computation of general Gauss sums to that of g1(χ). The absolute valueMathworldPlanetmathPlanetmathPlanetmath of g1(χ) is always p as long as χ is nontrivial, and if χ is a quadratic character (that is, χ(t) is the Legendre symbolMathworldPlanetmath (tp)), then the value of the Gauss sum is known to be



  • 1 Kenneth Ireland & Michael Rosen, A Classical Introduction to Modern Number TheoryMathworldPlanetmath, Second Edition, Springer–Verlag, 1990.
Title Gauss sum
Canonical name GaussSum
Date of creation 2013-03-22 12:48:28
Last modified on 2013-03-22 12:48:28
Owner djao (24)
Last modified by djao (24)
Numerical id 7
Author djao (24)
Entry type Definition
Classification msc 11L05
Related topic KloostermanSum