In mathematics, a Gauss sum or Gaussian sum is a particular kind of finite sum of roots of unity, typically $$G(\chi) := G(\chi, \psi)= \sum \chi(r)\cdot \psi(r)$$ where the sum is over elements $r$ of some finite commutative ring $R$, $ψ(r)$ is a group homomorphism of the additive group $R^+$ into the unit circle, and $χ(r)$ 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.