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

TL;DR

This paper investigates when polynomials of the form $b(x^d)$, derived from irreducible polynomials over finite fields, remain irreducible, providing necessary conditions and probability estimates.

Contribution

It establishes necessary conditions for the irreducibility of $b(x^d)$ and calculates the probability of irreducibility when these conditions are met.

Findings

Necessary conditions for irreducibility of $b(x^d)$ derived

Probability estimates for irreducibility provided

Conditions depend on properties of $b(x)$ and $d$

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.