Irreducibility of polynomials with square coefficients over finite fields

Abstract

We study a random polynomial of degree n over the finite field $F_q$, where the coefficients are independent and identically distributed and uniformly chosen from the squares in $F_q$. Our main result demonstrates that the likelihood of such a polynomial being irreducible approaches $1/n+O\left(q^{-1/2}\right)$ as the field size q grows infinitely large. The analysis we employ also applies to polynomials with coefficients selected from other specific sets.