  Given a polynomial with integral coefficients, one can inquire about the
possible residues it can take in its image modulo a prime $p$. The sum over the
distinct residues can sometimes be computed independent of the prime $p$; for
example, Gauss showed that the sum over quadratic residues vanishes modulo a
prime. In this paper we provide a closed form for the sum over distinct
residues in the image of Dickson polynomials of arbitrary degree over finite
fields of odd characteristic, and prove a complete characterization of the size
of the image set. Our result provides the first non-trivial classification of
such a sum for a family of polynomials of unbounded degree.
