On the 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 q, we denote the finite field of order q and by qm its extension of degree m. In this paper, we investigate the existence of a primitive normal pair (α,f(α)), with f(x) = ax2+bx+c dx+e qm(x) where the rank of the matrix F = abc 0 d e M2×3(qm) is 2. Namely, we establish sufficient conditions to show that nearly all fields of even characteristic possess such elements, except for 1100 1 0 if q = 2 and m is odd, and then we provide an explicit small list of possible and genuine exceptional pairs (q,m). © 2022 World Scientific Publishing Company.