On Irreducible Polynomials of the Form $b(x^d)$

Abstract

Let $p$ be a prime and $b(x)$ be an irreducible polynomial of degree $k$ over ${\mathbb{F}}_p$. Let $d\ge 1$ be an integer. Consider the following question: Is $b(x^d)$ irreducible? We derive necessary conditions for $b(x^d)$ to be irreducible. Further, when the necessary conditions are satisfied, we obtain the probability for $b(x^d)$ to be irreducible.