Primitive elements of finite fields $\mathbf{F}_{q^r}$ avoiding affine hyperplanes for $q=4$ and $q=5$
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866910331269283840 |
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| author | Grzywaczyk, Philipp Alexander Winterhof, Arne |
| author_facet | Grzywaczyk, Philipp Alexander Winterhof, Arne |
| contents | For a finite field $\mathbf{F}_{q^r}$ with fixed $q$ and $r$ sufficiently large, we prove the existence of a primitive element outside of a set of $r$ many affine hyperplanes for $q=4$ and $q=5$. This complements earlier results by Fernandes and Reis for $q\ge 7$. For $q=3$ the analogous result can be derived from a very recent bound on character sums of Iyer and Shparlinski. For $q=2$ the set consists only of a single element, and such a result is thus not possible. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_09192 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Primitive elements of finite fields $\mathbf{F}_{q^r}$ avoiding affine hyperplanes for $q=4$ and $q=5$ Grzywaczyk, Philipp Alexander Winterhof, Arne Number Theory 11T30, 11T24 For a finite field $\mathbf{F}_{q^r}$ with fixed $q$ and $r$ sufficiently large, we prove the existence of a primitive element outside of a set of $r$ many affine hyperplanes for $q=4$ and $q=5$. This complements earlier results by Fernandes and Reis for $q\ge 7$. For $q=3$ the analogous result can be derived from a very recent bound on character sums of Iyer and Shparlinski. For $q=2$ the set consists only of a single element, and such a result is thus not possible. |
| title | Primitive elements of finite fields $\mathbf{F}_{q^r}$ avoiding affine hyperplanes for $q=4$ and $q=5$ |
| topic | Number Theory 11T30, 11T24 |
| url | https://arxiv.org/abs/2402.09192 |