Primitive normal pairs with prescribed traces over finite fields
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913583736029184 |
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| author | Nath, Shikhamoni Mazumder, Arpan Chandra Basnet, Dhiren Kumar |
| author_facet | Nath, Shikhamoni Mazumder, Arpan Chandra Basnet, Dhiren Kumar |
| contents | Let $q$ be a positive integral power of some prime $p$ and $\mathbb{F}_{q^m}$ be a finite field with $q^m$ elements for some $m \in \mathbb{N}$. Here we establish a sufficient condition for the existence of primitive normal pairs of the type $(ε, f(ε))$ in $\mathbb{F}_{q^m}$ over $\mathbb{F}_{q}$ with two prescribed traces, $Tr_{{\mathbb{F}_{q^m}}/{\mathbb{F}_q}}(ε)=a$ and $Tr_{{\mathbb{F}_{q^m}}/{\mathbb{F}_q}}(f(ε))=b$, where $f(x) \in \mathbb{F}_{q^m}(x)$ is a rational function with some restrictions and $a, b \in \mathbb{F}_q$. Furthermore, for $q=5^k$, $m \geq 9$ and rational functions with degree sum 4, we explicitly find at most 12 fields in which the desired pair may not exist. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_19068 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Primitive normal pairs with prescribed traces over finite fields Nath, Shikhamoni Mazumder, Arpan Chandra Basnet, Dhiren Kumar Number Theory 12E20, 11T23 Let $q$ be a positive integral power of some prime $p$ and $\mathbb{F}_{q^m}$ be a finite field with $q^m$ elements for some $m \in \mathbb{N}$. Here we establish a sufficient condition for the existence of primitive normal pairs of the type $(ε, f(ε))$ in $\mathbb{F}_{q^m}$ over $\mathbb{F}_{q}$ with two prescribed traces, $Tr_{{\mathbb{F}_{q^m}}/{\mathbb{F}_q}}(ε)=a$ and $Tr_{{\mathbb{F}_{q^m}}/{\mathbb{F}_q}}(f(ε))=b$, where $f(x) \in \mathbb{F}_{q^m}(x)$ is a rational function with some restrictions and $a, b \in \mathbb{F}_q$. Furthermore, for $q=5^k$, $m \geq 9$ and rational functions with degree sum 4, we explicitly find at most 12 fields in which the desired pair may not exist. |
| title | Primitive normal pairs with prescribed traces over finite fields |
| topic | Number Theory 12E20, 11T23 |
| url | https://arxiv.org/abs/2405.19068 |