Primitive elements with zero traces

Let F-q 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 F-q(n) which is such that it and its reciprocal both have zero trace over F-q. Trivially, there is no such element when n < 5: we establish existence for all pairs (q, n) (n <greater than or equal to> 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 F-q whose coefficients of x and of x(n-1) are both zero. The method employs Kloosterman sums and a sieving technique. (C) 2000 Academic Press.