Existence of Primitive Polynomials with Three Coefficients Prescribed

Let Fq denote the finite field of q elements, q = p r for prime p and positive integer r. A monic polynomial f(x) = x+ ∑n i=1 fix n−i ∈ Fq[x] is called a primitive polynomial if it is irreducible over Fq and any of the roots of f can be used to generate the multiplicative group Fqn of Fqn . Equivalently, f is primitive if the smallest positive integer w such that f(x) | x − 1 is w = q − 1. Primitive polynomials and their roots are of interest in various applications of finite fields to coding theory and cryptography, and so it is of interest to know whether for a given q and n there exists a primitive polynomial of degree n over Fq which may satisfy certain additional conditions. One such condition is whether there exists a primitive polynomial of degree n over Fq with first coefficient f1 prescribed, where we note that f1 = −Tr(α), α a root of f and Tr the trace function from Fqn to Fq. This question has been answered (see [2], [6]), with answer as given in Theorem 1.1.