We prove congruences modulo 4 and modulo 8 for certain polynomial character sums, and use these congruences to give conditions for the nonvanishing of Brewer sums.

Brewer sums, Character sums, Dickson polynomials, Factorization of polynomials, Finite fields, Gauss sums, Jacobsthal sums, Special polynomials
dx.doi.org/10.1016/j.ffa.2005.06.002
Finite Fields and their Applications
School of Mathematics and Statistics

Alaca, S. (2007). Congruences for Brewer sums. Finite Fields and their Applications, 13(1), 1–19. doi:10.1016/j.ffa.2005.06.002