Eisenstein divisibility

The heart of “Eisenstein divisibility” is just the three divisibility checks in Eisenstein’s Criterion for a polynomial

$f(x)=a_n{x}^n+a_{n-1}{x}^{n-1}+\cdots +a_{1}x+a_0\in \mathbb{Z}[x]$ Pick a prime $p$ and verify:

1. $p$ does not divide the leading coefficient $p\nmid a_n$.
2. $p$ does divide every other coefficient $p\mid a_{n-1},p\mid a_{n-2},\dots ,p\mid a_0$
3. $p^2$ does not divide the constant term $p^2\nmid a_0$

If you can find such a prime $p$, then Eisenstein’s Criterion tells you immediately that $f(x)$ is irreducible over $\mathbb{Q}$.