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 $d\geq 1$ 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.