On existence of primitive normal elements of rational form over finite fields of even characteristic
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arXiv
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| Format: | Preprint |
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2020
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| _version_ | 1866929582321434624 |
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| author | Hazarika, Himangshu Basnet, Dhiren Kumar Kapetanakis, Giorgos |
| author_facet | Hazarika, Himangshu Basnet, Dhiren Kumar Kapetanakis, Giorgos |
| contents | 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 $(q,m)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2005_01216 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | On existence of primitive normal elements of rational form over finite fields of even characteristic Hazarika, Himangshu Basnet, Dhiren Kumar Kapetanakis, Giorgos Number Theory 12E20, 11T23 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 $(q,m)$. |
| title | On existence of primitive normal elements of rational form over finite fields of even characteristic |
| topic | Number Theory 12E20, 11T23 |
| url | https://arxiv.org/abs/2005.01216 |