Existence of rational primitive normal pairs over finite fields

Abstract. For a finite field Fqn and a rational function f = f1 f2 ∈ Fqn(x), we present a sufficient condition for the existence of a primitive normal element α ∈ Fqn in such a way f(α) is also primitive in Fqn, where f(x) is a rational function in Fqn(x) of degree sum m (degree sum of f(x) = f1(x) f2(x) is defined to be the sum of the degrees of f1(x) and f2(x)). Additionally, for rational functions of degree sum 4, we proved that there are only 37 and 16 exceptional values of (q, n) when q = 2k and q = 3k respectively