Primitive normal Values of rational functions with one prescribed norm and trace over finite fields
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| Format: | Preprint |
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2024
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| _version_ | 1866910439277854720 |
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| author | Mazumder, Arpan Chandra Basnet, Dhiren Kumar |
| author_facet | Mazumder, Arpan Chandra Basnet, Dhiren Kumar |
| contents | Let $q, n, m \in \mathbb{N}$ be such that $q$ is a prime power and $a, b \in \mathbb{F}$. In this article we establish a sufficient condition for the existence of a primitive normal pair $(α, f(α)) \in \mathbb{F}_{q^m}$ over $\mathbb{F}$ with a prescribed primitive norm $a$ and a non-zero trace $b$ over $\mathbb{F}$ of $α$, where $f(x) \in \mathbb{F}_{q^m}(x)$ is a rational function of degree sum $n$ with some minor restrictions. Furthermore, for $q=7^k$, $m \geq 7$ and rational functions with numerator and denominator being linear, we explicitly find at most 6 fields in which the desired pair may not exist. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_05298 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Primitive normal Values of rational functions with one prescribed norm and trace over finite fields Mazumder, Arpan Chandra Basnet, Dhiren Kumar Number Theory 12E20, 11T23 Let $q, n, m \in \mathbb{N}$ be such that $q$ is a prime power and $a, b \in \mathbb{F}$. In this article we establish a sufficient condition for the existence of a primitive normal pair $(α, f(α)) \in \mathbb{F}_{q^m}$ over $\mathbb{F}$ with a prescribed primitive norm $a$ and a non-zero trace $b$ over $\mathbb{F}$ of $α$, where $f(x) \in \mathbb{F}_{q^m}(x)$ is a rational function of degree sum $n$ with some minor restrictions. Furthermore, for $q=7^k$, $m \geq 7$ and rational functions with numerator and denominator being linear, we explicitly find at most 6 fields in which the desired pair may not exist. |
| title | Primitive normal Values of rational functions with one prescribed norm and trace over finite fields |
| topic | Number Theory 12E20, 11T23 |
| url | https://arxiv.org/abs/2405.05298 |