Existence of primitive k-normal elements for critical values over finite fields
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| Format: | Preprint |
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2025
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| _version_ | 1866917052501983232 |
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| author | Aguirre, Josimar J. R. Mazzini, Sarah F. M. Neumann, Victor G. L. |
| author_facet | Aguirre, Josimar J. R. Mazzini, Sarah F. M. Neumann, Victor G. L. |
| contents | Let $\mathbb{F}_{q^n}$ be a finite field with $q^n$ elements. An element $α\in \mathbb{F}_{q^n}$ is called $k$-normal over $\mathbb{F}_q$ if $α$ and its conjugates generate a vector subspace of $\mathbb{F}_{q^n}$ of dimension $n-k$ over $\mathbb{F}_q$. The existence of primitive $k$-normal elements and related properties have been studied throughout the past few years for $k > n/2$. In this paper, we provide general results on the existence of primitive $k$-normal elements for the critical value $k = n/2$, which have not been studied until now, except for $n = 4$. Furthermore, we show the strength of this result by providing a complete characterization of the existence of primitive $3$-normal elements in $\mathbb{F}_{q^6}$ over $\mathbb{F}_q$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_26972 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Existence of primitive k-normal elements for critical values over finite fields Aguirre, Josimar J. R. Mazzini, Sarah F. M. Neumann, Victor G. L. Number Theory 12E20, 11T30 Let $\mathbb{F}_{q^n}$ be a finite field with $q^n$ elements. An element $α\in \mathbb{F}_{q^n}$ is called $k$-normal over $\mathbb{F}_q$ if $α$ and its conjugates generate a vector subspace of $\mathbb{F}_{q^n}$ of dimension $n-k$ over $\mathbb{F}_q$. The existence of primitive $k$-normal elements and related properties have been studied throughout the past few years for $k > n/2$. In this paper, we provide general results on the existence of primitive $k$-normal elements for the critical value $k = n/2$, which have not been studied until now, except for $n = 4$. Furthermore, we show the strength of this result by providing a complete characterization of the existence of primitive $3$-normal elements in $\mathbb{F}_{q^6}$ over $\mathbb{F}_q$. |
| title | Existence of primitive k-normal elements for critical values over finite fields |
| topic | Number Theory 12E20, 11T30 |
| url | https://arxiv.org/abs/2510.26972 |