: 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 5) expact (4,5),(2,6),and (3,6).