Primitive element pairs with a prescribed trace in the quartic extension of a finite field

In this paper, we give a largely self-contained proof that the quartic extension [Formula: see text] of the finite field [Formula: see text] contains a primitive element [Formula: see text] such that the element [Formula: see text] is also a primitive element of [Formula: see text] and [Formula: see text] for any prescribed [Formula: see text]. The corresponding result has already been established for finite field extensions of degrees exceeding 4 in [Primitive element pairs with one prescribed trace over a finite field, Finite Fields Appl. 54 (2018) 1–14.].