A special type of primitive normal pair in a finite field of odd characteristic

Let [Formula: see text] with [Formula: see text] an odd prime power, and let [Formula: see text] denote the degree-[Formula: see text] extension of the finite field [Formula: see text]. In this paper, we study the existence of an element [Formula: see text] for which both [Formula: see text] and [Formula: see text] are simultaneously primitive and normal over [Formula: see text]. Furthermore, when [Formula: see text] for some [Formula: see text], such a pair is guaranteed except for exact 3 choices of [Formula: see text].