  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.
