References & Citations
Mathematics > Number Theory
Title: On existence of primitive normal elements of rational form over finite fields of even characteristic
(Submitted on 4 May 2020 (v1), last revised 26 Aug 2021 (this version, v2))
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 $(\alpha, \, f(\alpha))$, 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 $(q,m)$.
Submission history
From: Dhiren Kumar Basnet [view email][v1] Mon, 4 May 2020 00:09:32 GMT (21kb)
[v2] Thu, 26 Aug 2021 01:15:56 GMT (24kb)
Link back to: arXiv, form interface, contact.