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.

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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