An irreducibility criterion for integer polynomials

TL;DR

This paper establishes a new irreducibility criterion for integer polynomials based on coefficient inequalities and prime evaluations at certain integers, providing a practical test for polynomial irreducibility.

Contribution

It introduces a novel irreducibility criterion for integer polynomials involving coefficient orderings and prime evaluations, extending existing methods.

Findings

If the leading coefficient or polynomial value at a certain integer is prime, the polynomial is irreducible.

The criterion applies to polynomials with ordered coefficients or dominant leading term.

Provides a practical test for irreducibility based on prime number conditions.

Abstract

Let $f(x) = \sum\limits _{i=0}^{n} a_i x^i $ be a polynomial with coefficients from the ring $\mathbb{Z}$ of integers satisfying either $(i)$ $0 < a_0 \leq a_{1} \leq \cdots \leq a_{k-1} < a_{k} < a_{k+1} \leq \cdots \leq a_n$ for some $k$, $0 \leq k \leq n-1$; or $(ii)$ $|a_n| > |a_{n-1}| + \cdots + |a_{0}|$ with $a_0 \neq 0$. In this paper, it is proved that if $|a_n|$ or $|f(m)|$ is a prime number for some integer $m$ with $|m|\geq 2 $ then the polynomial $f(x)$ is irreducible over $\mathbb{Z}$.