Revisiting Eisenstein-type criterion over integers

TL;DR

This paper provides an accessible, elementary proof for a generalization of Eisenstein's irreducibility criterion over integers, extending it to cases where the divisibility conditions involve higher powers of a prime.

Contribution

It offers a simplified, undergraduate-friendly proof of a generalized Eisenstein criterion for specific cases of k, using basic divisibility arguments.

Findings

The criterion holds for k=2,3,4 with elementary proofs.

The proof relies on basic divisibility properties of integers.

The result extends Eisenstein's criterion to higher powers of primes.

Abstract

The following result, a consequence of Dumas criterion for irreducibility of polynomials over integers, is generally proved using the notion of Newton diagram: Let $f(x)$ be a polynomial with integer coefficients and $k$ be a positive integer relatively prime to the degree of $f(x)$. Suppose that there exists a prime number $p$ such that the leading coefficient of $f(x)$ is not divisible by $p$, all the remaining coefficients are divisible by $p^k$, and the constant term of $f(x)$ is not divisible by $p^{k+1}$. Then $f(x)$ is irreducible over $\mathbb{Z}$. For $k=1$, this is precisely the Eisenstein criterion. The aim of this article is to give an alternate proof, accessible to the undergraduate students, of this result for $k\in \{2,3,4\}$ using basic divisibility properties of integers.