Primitive elements and polynomials with arbitrary trace (Q922593)

Let \(F_ q\) denote the finite field of order q. For \(\xi \in F_{q^ n}\), the trace T(\(\xi\)) of \(\xi\) over \(F_ q\) is given by \(T(\xi)=\xi +\xi^ q+\xi^{q^ 2}+...+\xi^{q^{n-1}}.\) The author shows that except for necessary exceptions, in any proper extension field \(F_{q^ n}\) over \(F_ q\), there is a primitive element with arbitrary trace in \(F_ q\). More specifically he proves: Let \(n\geq 2\) and t be an arbitrary element of \(F_ q\) with \(t\neq 0\) if \(n=2\) or if \(n=3\) and \(q=4\). Then there exists a primitive element in \(F_{q^ n}\) with trace t. Equivalently, there exits a primitive polynomial of degree n over \(F_ q\) with trace t. The author's method relies upon obtaining estimates for certain character sums. His result greatly extends those of \textit{O. Moreno} [ibid. 41, 53-56 (1982; Zbl 0485.12015)] and [J. Comb. Theory, Ser. A 51, 104-110 (1989; Zbl 0682.12015)] which consider the cases of primitive elements of trace 1 over \(F_{2^ n}\) and primitive quadratics of trace 1 over \(F_{q^ n}\).