Irreducibility of Polynomials with Square Coefficients over Finite Fields
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866910660492787712 |
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| author | Bary-Soroker, Lior Shmueli, Roy |
| author_facet | Bary-Soroker, Lior Shmueli, Roy |
| contents | We study a random polynomial of degree $n$ over the finite field $\mathbb{F}_q$, where the coefficients are independent and identically distributed and uniformly chosen from the squares in $\mathbb{F}_q$. Our main result demonstrates that the likelihood of such a polynomial being irreducible approaches $1/n + O(q^{-1/2})$ as the field size $q$ grows infinitely large. The analysis we employ also applies to polynomials with coefficients selected from other specific sets. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_16814 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Irreducibility of Polynomials with Square Coefficients over Finite Fields Bary-Soroker, Lior Shmueli, Roy Number Theory 11T06, 11R09, 12E05, 12E20 We study a random polynomial of degree $n$ over the finite field $\mathbb{F}_q$, where the coefficients are independent and identically distributed and uniformly chosen from the squares in $\mathbb{F}_q$. Our main result demonstrates that the likelihood of such a polynomial being irreducible approaches $1/n + O(q^{-1/2})$ as the field size $q$ grows infinitely large. The analysis we employ also applies to polynomials with coefficients selected from other specific sets. |
| title | Irreducibility of Polynomials with Square Coefficients over Finite Fields |
| topic | Number Theory 11T06, 11R09, 12E05, 12E20 |
| url | https://arxiv.org/abs/2410.16814 |