In algebraic number theory, a Gauss sum or Gaussian sum is a particular kind of finite sum of roots of unity, typically G := G = ∑ χ ⋅ ψ {\displaystyle G:=G=\sum \chi \cdot \psi } 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. Wikipedia