Congruences for Brewer sums (Q865377)

The Brewer sums considered here are \[ \Lambda_n(a)=\sum_{x=0}^{p-1} L(D_n(x,a),p), \] where \(p\) is a prime, \(L\) is the Legendre symbol and \(D_n(x,a)\) is the \(n\)th order Dickson polynomial of the first kind. \(\Lambda_n(a)\) is easily seen to be 0 when \((n, p^2-1)=1\) or \(p\equiv 3\pmod{4}\). The author considers the case where \(p\equiv 1\pmod{4}\), \(n\) is an odd prime dividing \(p^2-1\) and determines which \(\Lambda_n(a)\) are 0. The proof uses explicit factorizations of the Dickson polynomials over finite fields.