On the existence of primitive normal elements of rational form over finite fields of even characteristic

Abstract

Let [Formula: see text] be an even prime power and [Formula: see text] an integer. By [Formula: see text], we denote the finite field of order [Formula: see text] and by [Formula: see text] its extension of degree [Formula: see text]. In this paper, we investigate the existence of a primitive normal pair [Formula: see text], with [Formula: see text] where the rank of the matrix [Formula: see text] is 2. Namely, we establish sufficient conditions to show that nearly all fields of even characteristic possess such elements, except for [Formula: see text] if [Formula: see text] and [Formula: see text] is odd, and then we provide an explicit small list of possible and genuine exceptional pairs [Formula: see text].