On primitive polynomials over finite fields

Abstract

Let F= GF( q) and let E = GF( q k ) 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 T E F (ω) = a . Equivalently, the coefficient of x k − 1 in a primitive polynomial of degree k over GF( q) may be arbitrarily selected from GF(q) ∗ .