= Eisenstein criterion
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= Eisenstein
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= Eisenstein's criterion
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For a primitive polynomial
$$
F(X)=a_nX^n+\cdots+a_0\in\mathbb Z[X],
$$
if a prime $p$ does not divide $a_n$, divides every $a_j$ for $j<n$, and has $p^2\nmid a_0$, then $F$ is irreducible. If $F=GH$, reduction modulo $p$ forces both $\overline G$ and $\overline H$ to be positive-degree monomials. Their constant terms are consequently divisible by $p$, making $p^2\mid F(0)$, a contradiction. <Gauss lemma for polynomials> transfers the result between $\mathbb Z[X]$ and $\mathbb Q[X]$.
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