Irreducibility of Polynomials with Square Coefficients over Finite Fields

TL;DR

This paper investigates the probability that a random polynomial over a finite field, with coefficients chosen uniformly from squares, is irreducible, showing it approaches 1/n as the field size increases.

Contribution

It provides the first asymptotic probability result for irreducibility of polynomials with square coefficients over finite fields.

Findings

Probability of irreducibility approaches 1/n as q increases

Method applies to polynomials with coefficients from other specific sets

Asymptotic behavior characterized for large finite fields

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.