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arXiv · 2410.16814

Irreducibility of Polynomials with Square Coefficients over Finite Fields

Lior Bary-Soroker·Roy Shmueli
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Abstract

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.

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Lior Bary-Soroker, Roy Shmueli. 2024-10-22. Irreducibility of Polynomials with Square Coefficients over Finite Fields. https://arxiv.org/abs/2410.16814

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