# An irreducibility criterion for polynomials over integers

## TL;DR

This paper introduces a new criterion for determining the irreducibility of certain integer polynomials, showing they have cyclotomic factors if reducible, and provides practical methods for checking irreducibility and separability.

## Contribution

It establishes a novel irreducibility criterion for polynomials with specific coefficient conditions and offers practical procedures for testing irreducibility and separability.

## Key findings

- Polynomials with prime constant term and sum of coefficients have cyclotomic factors if reducible.
- A simple irreducibility check for certain trinomials is provided.
- Separability criteria for specific quadrinomials are established.

## Abstract

In this article, we consider the polynomials of the form $f(x)=a_0+a_1x+a_2x^2+\cdots+a_nx^n\in \mathbb{Z}[x],$ where $|a_0|=|a_1|+\dots+|a_n|$ and $|a_0|$ is a prime. We show that these polynomials have a cyclotomic factor whenever reducible. As a consequence, we give a simple procedure for checking the irreducibility of trinomials of this form and separability criterion for certain quadrinomials.