Primitive Elements with Zero Traces,

Let Fq denote the finite field of order q, a power of a prime p, and n be a positive integer. We resolve completely the question of whether there exists a primitive element of Fqn which is such that it and its reciprocal both have zero trace over Fq. Trivially, there is no such element when n<5: we establish existence for all pairs (q, n) (n≥5) except (4, 5), (2, 6), and (3, 6). Equivalently, with the same exceptions, there is always a primitive polynomial P(x) of degree n over Fq whose coefficients of x and of xn-1 are both zero. The method employs Kloosterman sums and a sieving technique.