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| Natura: | Preprint |
| Pubblicazione: |
2025
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| Accesso online: | https://arxiv.org/abs/2507.21515 |
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| _version_ | 1866909710381219840 |
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| author | Bagger, Gustav Kjærbye Punch, James |
| author_facet | Bagger, Gustav Kjærbye Punch, James |
| contents | Let $r \geq 2$ be an integer, $q$ a prime power and $\mathbb{F}_{q}$ the finite field with $q$ elements. Consider the problem of showing existence of primitive elements in a subset $\mathcal{A} \subseteq \mathbb{F}_{q^r}$. We prove a sieve criterion for existence of such elements, dependent only on an estimate for the character sum $\sum_{γ\in \mathcal{A}}χ(γ)$. The flexibility and direct applicability of our criterion should be of considerable interest for problems in this field. We demonstrate the utility of our result by tackling a problem of Fernandes and Reis (2021) with $\mathcal{A}$ avoiding affine hyperplanes, obtaining significant improvements over previous knowledge. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_21515 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The modified prime sieve for primitive elements in finite fields Bagger, Gustav Kjærbye Punch, James Number Theory 11T24, 12E20 Let $r \geq 2$ be an integer, $q$ a prime power and $\mathbb{F}_{q}$ the finite field with $q$ elements. Consider the problem of showing existence of primitive elements in a subset $\mathcal{A} \subseteq \mathbb{F}_{q^r}$. We prove a sieve criterion for existence of such elements, dependent only on an estimate for the character sum $\sum_{γ\in \mathcal{A}}χ(γ)$. The flexibility and direct applicability of our criterion should be of considerable interest for problems in this field. We demonstrate the utility of our result by tackling a problem of Fernandes and Reis (2021) with $\mathcal{A}$ avoiding affine hyperplanes, obtaining significant improvements over previous knowledge. |
| title | The modified prime sieve for primitive elements in finite fields |
| topic | Number Theory 11T24, 12E20 |
| url | https://arxiv.org/abs/2507.21515 |