What type of study is this?

September 10, 2025Open Access

On the Irreducibility of Polynomials with Prime Power Shifts

Key Points

The polynomial f(X) + p^k g(X) is mostly irreducible over Q or factors into polynomials with degree multiples of gcd(k, deg g).
The findings show that for all but finitely many primes p, no conditions on gcd(k, deg g) are necessary for irreducibility.
This research builds on prior work on polynomial irreducibility, generalizing results to a broader class of polynomials.
The exploration of irreducibility is essential in understanding polynomial behavior across different prime powers and coefficients.

Abstract

In this paper, we study the irreducibility of polynomials of the form \ (f (X) + pᵏ 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 \ ( (k, g) \). We prove that, for all but finitely many primes \ (p \), the polynomial \ (f (X) + pᵏ g (X) \) is either irreducible over \ (Q \) or factors into polynomials whose degrees are multiples of \ ( (k, g) \). This generalizes and extends earlier work on the irreducibility of such polynomials.

Read Full Paperexternally

AI에게 질문

Bookmark

View Full Paper