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

Abstract

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).

On the existence of pairs of primitive and normal elements over finite fields

“…8 we get A t = 1380.449 and (6) is satisfied for n ≥ 127. Using SageMath ([12]) we test condition 5 n/2 ≥ 6W (5 n − 1) 2 W 5 (x n − 1) for n < 127 and we get that it holds for all n ≥ 25 and for n ∈ {11, 13, 15, 17, 19, 20, 21, 22, 23}.…”