A primitive normal pair in a finite field with prescribed traces and norms
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| Format: | Preprint |
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2023
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| author | Chatterjee, Kaustav Sharma, Hariom Tiwari, Shailesh Kumar |
| author_facet | Chatterjee, Kaustav Sharma, Hariom Tiwari, Shailesh Kumar |
| contents | Given ${\mathbb{F}_{p^t}}$, a field with $p^t$ elements, where $p$ is a prime power, $t$ is a positive integer. Let $f(x)$ be a polynomial over $\mathbb{F}_{p^t}$ of degree $m$ with some restrictions. In this paper, we construct a sufficient condition on $(p,t)$ which guarantees the existence of a primitive normal pair $(ε,f(ε))$ such that $Tr_{\mathbb{F}_{p^t}/\mathbb{F}_p}(ε)=a$, $Tr_{\mathbb{F}_{p^t}/\mathbb{F}_p}(f(ε))=b$ and $N_{\mathbb{F}_{p^t}/\mathbb{F}_p}(ε)=c$, $N_{\mathbb{F}_{p^t}/\mathbb{F}_p}(f(ε))=d$ where $c,d\in\mathbb{F}_{p}$ are primitive elements and $a,b\in\mathbb{F}_{p}^*$. Furthermore, we demonstrate that, for $p=11^k;$ $k\geq1,$ $m=8$ and $t\geq 15$, there are only $4$ possible exceptions where such pairs may not exist. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2308_02846 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A primitive normal pair in a finite field with prescribed traces and norms Chatterjee, Kaustav Sharma, Hariom Tiwari, Shailesh Kumar Number Theory 12E20-Finite fields (field-theoretic aspects), 11T23-Exponential sums Given ${\mathbb{F}_{p^t}}$, a field with $p^t$ elements, where $p$ is a prime power, $t$ is a positive integer. Let $f(x)$ be a polynomial over $\mathbb{F}_{p^t}$ of degree $m$ with some restrictions. In this paper, we construct a sufficient condition on $(p,t)$ which guarantees the existence of a primitive normal pair $(ε,f(ε))$ such that $Tr_{\mathbb{F}_{p^t}/\mathbb{F}_p}(ε)=a$, $Tr_{\mathbb{F}_{p^t}/\mathbb{F}_p}(f(ε))=b$ and $N_{\mathbb{F}_{p^t}/\mathbb{F}_p}(ε)=c$, $N_{\mathbb{F}_{p^t}/\mathbb{F}_p}(f(ε))=d$ where $c,d\in\mathbb{F}_{p}$ are primitive elements and $a,b\in\mathbb{F}_{p}^*$. Furthermore, we demonstrate that, for $p=11^k;$ $k\geq1,$ $m=8$ and $t\geq 15$, there are only $4$ possible exceptions where such pairs may not exist. |
| title | A primitive normal pair in a finite field with prescribed traces and norms |
| topic | Number Theory 12E20-Finite fields (field-theoretic aspects), 11T23-Exponential sums |
| url | https://arxiv.org/abs/2308.02846 |