For [Formula: see text] ([Formula: see text]), denote by [Formula: see text] the finite field of order [Formula: see text] and for a positive integer [Formula: see text], let [Formula: see text] be its extension field of degree [Formula: see text]. We establish a sufficient condition for existence of a primitive normal element [Formula: see text] such that [Formula: see text] is a primitive element, where [Formula: see text], with [Formula: see text] satisfying [Formula: see text] in [Formula: see text].

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].

For q = 3 r (r > 0), denote by F q the finite field of order q and for a positive integer m ≥ 2, let F q m be its extension field of degree m. We establish a sufficient condition for existence of a primitive normal element α such that f (α) is a primitive element, where f (x) = ax 2 + bx + c, with a, b, c ∈ F q m satisfying b 2 = ac in F q m except for at most 9 exceptional pairs (q, m).

Let q be an even prime power and m ≥ 2 an integer. By F q , we denote the finite field of order q and by F q m , its extension degree m. In this paper we investigate the existence of primitive normal pair (α, f (α)), where f (x) = ax 2 +bx+c dx+e ∈ F q m (x) with a = 0, dx + e = 0 in F q m and establish some sufficient conditions to show that nearly all fields of even characteristic possess such elements. We conclude the paper by providing an explicit small list of genuine exceptional pairs (q, m).