Irreducibility of polynomials with square coefficients over finite fields

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 ( 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.