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

Abstract

Let $q$ be an even prime power and $m\geq2$ an integer. By $\mathbb{F}_q$, we denote the finite field of order $q$ and by $\mathbb{F}_{q^m}$ its extension degree $m$. In this paper we investigate the existence of a primitive normal pair $(α, \, f(α))$, with $f(x)= \dfrac{ax^2+bx+c}{dx+e} \in \mathbb{F}_{q^m}(x)$, where the rank of the matrix $F= \begin{pmatrix}a \, &b\, & c\\ 0\, &d \, &e \end{pmatrix}$ $\in M_{2 \times 3}(\Fm) $ is 2. Namely, we establish sufficient conditions to show that nearly all fields of even characteristic possess such elements, except for $\begin{pmatrix} 1 \, &1 \, & 0\\ 0\, &1 \, &0 \end{pmatrix}$ if $q=2$ and $m$ is odd, and then we provide an explicit list of possible and genuine exceptional pairs $(q,m)$.

Theorems & Definitions (39)

• Theorem 1.1: Primitive normal basis theorem, 3
• Theorem 1.2: Strong primitive normal basis theorem 4
• Theorem 1.3: 27
• Theorem 1.4: 8
• Theorem 1.5: 1
• Theorem 1.6
• Theorem 1.7
• Definition 2.1
• Lemma 2.1: 10, Theorem 5.4 - Orthogonality relations
• Lemma 2.2: 6, Corollary 2.3