A primitive normal pair with prescribed prenorm
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866910501936562176 |
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| author | Chatterjee, K. Tiwari, S. K. |
| author_facet | Chatterjee, K. Tiwari, S. K. |
| contents | For any positive integers $q$, $n$, $m$ with $q$ being a prime power and $n \geq 5$, we establish a condition sufficient to ensure the existence of a primitive normal pair $(ε,f(ε))$ in $\mathbb{F}_{q^{n}}$ over $\mathbb{F}_{q}$ such that $\mathrm{PN}_{q^n/q}(ε)=a$, where $a\in\mathbb{F}_{q}$ is prescribed. Here $f={f_{1}}/{f_{2}}\in\mathbb{F}_{q^n}(x)$ is a rational function subject to some minor restrictions such that deg($f_{1}$)+deg($f_{2}$)$=m$ and $\mathrm{PN}_{q^n/q}(ε)
=\sum_{i=0}^{n-1}\Bigg(\underset{j\neq i}{\underset{0\leq j\leq n-1}{\prod_{}^{}}}ε^{q^j}\Bigg)$. Finally, we conclude that for $m=3$, $n\geq 6$, and $q=7^k$ where $k\in\mathbb{N}$, such a pair will exist certainly for all $(q,n)$ except possibly $10$ choices at most. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_03571 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A primitive normal pair with prescribed prenorm Chatterjee, K. Tiwari, S. K. Number Theory 11T23, 12E20 For any positive integers $q$, $n$, $m$ with $q$ being a prime power and $n \geq 5$, we establish a condition sufficient to ensure the existence of a primitive normal pair $(ε,f(ε))$ in $\mathbb{F}_{q^{n}}$ over $\mathbb{F}_{q}$ such that $\mathrm{PN}_{q^n/q}(ε)=a$, where $a\in\mathbb{F}_{q}$ is prescribed. Here $f={f_{1}}/{f_{2}}\in\mathbb{F}_{q^n}(x)$ is a rational function subject to some minor restrictions such that deg($f_{1}$)+deg($f_{2}$)$=m$ and $\mathrm{PN}_{q^n/q}(ε) =\sum_{i=0}^{n-1}\Bigg(\underset{j\neq i}{\underset{0\leq j\leq n-1}{\prod_{}^{}}}ε^{q^j}\Bigg)$. Finally, we conclude that for $m=3$, $n\geq 6$, and $q=7^k$ where $k\in\mathbb{N}$, such a pair will exist certainly for all $(q,n)$ except possibly $10$ choices at most. |
| title | A primitive normal pair with prescribed prenorm |
| topic | Number Theory 11T23, 12E20 |
| url | https://arxiv.org/abs/2406.03571 |