In algebraic number theory, a Gauss sum or Gaussian sum is a particular kind of finite sum of roots of unity, typically where the sum is over elements r of some finite commutative ring R, ψ is a group homomorphism of the additive group R+ into the unit circle, and χ is a group homomorphism of the unit group R× into the unit circle, extended to non-unit r, where it takes the value 0.
Gauss sums are the analogues for finite fields of the Gamma function.
The absolute value of Gauss sums is usually found as an application of Plancherel's theorem on finite groups.
In the case where R is a field of p elements and χ is nontrivial, the absolute value is p1⁄2.
The Gauss sum of a Dirichlet character modulo N is If χ is also primitive, then in particular, it is nonzero.