On the Existence of Pairs of Primitive and Normal Elements Over Finite Fields

Let $$\mathbb {F}_{q^n}$$ be a finite field with $$q^n$$ elements, and let $$m_1$$ $$m_2$$ positive integers. Given polynomials $$f_1(x), f_2(x) \in \mathbb {F}_{q^n}[x]$$ $$\deg (f_i(x)) \le m_i$$ , for $$i = 1, 2$$ such that the rational function $$f_1(x)/f_2(x)$$ satisfies certain conditions which we define, present sufficient condition existence of primitive element $$\alpha normal over {F}_q$$ $$f_1(\alpha )/f_2(\alpha )$$ is also primitive.