Title: On primitive polynomials over finite fields
Abstract: Let F=GF(q) and let E = GF(qk) be the field extension of degree k of F. We show that the following statement holds for all but finitely many exceptional pairs (q, k): Given any element aϵF∗ there exists a primitive element ω of E with trace TEF(ω) = a. Equivalently, the coefficient of xk − 1 in a primitive polynomial of degree k over GF(q) may be arbitrarily selected from GF(q)∗.