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

Abstract

Let $q$ be an even prime power and $m\ge 2$ 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 $(\alpha ,f(\alpha ))$, with $f(x)=\frac{a{x}^2+bx+c}{dx+e}\in {\mathbb{F}}_{q^m}(x)$, where the rank of the matrix $F=\left(\begin{array}{ccc}a& b& c\\ 0& d& e\end{array}\right)$ $\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 $\left(\begin{array}{ccc}1& 1& 0\\ 0& 1& 0\end{array}\right)$ if $q=2$ and $m$ is odd, and then we provide an explicit list of possible and genuine exceptional pairs $(q,m)$.

24 pages. arXiv admin note: text overlap with arXiv:2001.06977