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 exi sts 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 t o> 5) except (4, 5), (2, 6), and (3, 6). Equivalently, with the same except ions, there is always a primitive polynomial P(x) of degree n over F-q whos e coefficients of x and of x(n-1) are both zero. The method employs Klooste rman sums and a sieving technique. (C) 2000 Academic Press.