On the Irreducibility of Polynomials with Prime Power Shifts

Abstract

In this paper, we study the irreducibility of polynomials of the form \( f(X) + p^k g(X) \), where \( f(X) \) and \( g(X) \) are polynomials with integer coefficients, \( p \) is a prime number, and \( k \) is a positive integer. Unlike previous results, we do not require \( f(X) \) and \( g(X) \) to be relatively prime or impose any conditions on \( \gcd(k, \deg g) \). We prove that, for all but finitely many primes \( p \), the polynomial \( f(X) + p^k g(X) \) is either irreducible over \( \mathbb{Q} \) or factors into polynomials whose degrees are multiples of \( \gcd(k, \deg g) \). This generalizes and extends earlier work on the irreducibility of such polynomials.