Congruences for Brewer sums
by
Saban Alaca
Carleton University

For each natural number n and each nonzero complex number a, the Dickson polynomial V_n (x, a) of the first kind of degree n is defined by

Vn (x, a) = [n/2] å j=0  n n-j æ è n-j j ö ø (-1)j aj xn-2j .

For an odd prime p and an integer a not divisible by p, the generalized Brewer sum L_n (a) is defined for a natural number n by

Ln (a) = p-1 å x=0  æ è Vn (x, a) p ö ø ,

where ( [*/p] ) is the Legendre symbol mod p.

An important question is to determine a necessary and sufficient condition for L_n (a) = 0 (or L_n (a) ≠ 0). We deal with the question when n is an odd prime. It is known that V_n (x, a) is a permutation polynomial over F_p if and only if (n, p² - 1) = 1, and so L_n (a) = 0. For (n, p² -1) > 1, the question of determining when L_n (a) = 0 remains open.

If p ≡ 3 mod 4, then it is known that L_n (a) = 0. Thus, the question remains open when p ≡ 1 mod 4. For odd primes n and p with (n, p² -1) > 1 and p ≡ 1 mod 4, either p ≡ 1 mod n or p ≡ -1 mod n.

For these two cases, 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.

Let n and p be odd primes with (n, p² -1) > 1 and p ≡ 1 mod 4.

(i) Let p ≡ 1 mod n. If a ∈ F_p is a quadratic residue and n ≡ 3 \smod 4, or if a ∈ F_p is a quadratic nonresidue, p ≡ 5 mod 8 and n ≡ 3 mod 4, then L_n (a) ≠ 0.

(ii) Let p ≡ -1 mod n. If a ∈ F_p is a quadratic residue, p ≡ 5 mod 8 and n ≡ 3 mod 4, or if a ∈ F_p is a quadratic nonresidue and n ≡ 3 mod 4, then L_n (a) ≠ 0.

In order to prove these results, we establish some explicit results about the factorization of Dickson polynomials over F_p by using the work of Bhargava on the factorization of Dickson's polynomials.

Date received: June 15, 2008