Divisibility of the Multiplicative Order Modulo Monic Irreducible Polynomials Over Finite Fields
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866908600921751552 |
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| author | Da Conceição, Joaquim Cera |
| author_facet | Da Conceição, Joaquim Cera |
| contents | We consider the set of monic irreducible polynomials $P$ over a finite field $\mathbb{F}_q$ such that the multiplicative order modulo $P$ of some a in $\mathbb{F}_q(T)$ is divisible by a fixed positive integer $d$. Call $R_q(a,d)$ this set. We show the existence or non-existence of the density of $R_q(a,d)$ for three distinct notions of density. In particular, the sets $R_q(a,d)$ have a Dirichlet density. Under some assumptions, we prove simple formulas for the density values. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_09107 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Divisibility of the Multiplicative Order Modulo Monic Irreducible Polynomials Over Finite Fields Da Conceição, Joaquim Cera Number Theory 11R44, 11T06, 11N37, 11R58 We consider the set of monic irreducible polynomials $P$ over a finite field $\mathbb{F}_q$ such that the multiplicative order modulo $P$ of some a in $\mathbb{F}_q(T)$ is divisible by a fixed positive integer $d$. Call $R_q(a,d)$ this set. We show the existence or non-existence of the density of $R_q(a,d)$ for three distinct notions of density. In particular, the sets $R_q(a,d)$ have a Dirichlet density. Under some assumptions, we prove simple formulas for the density values. |
| title | Divisibility of the Multiplicative Order Modulo Monic Irreducible Polynomials Over Finite Fields |
| topic | Number Theory 11R44, 11T06, 11N37, 11R58 |
| url | https://arxiv.org/abs/2412.09107 |