Eisenstein’s Criterion

Let $f(x)=a_nx^n+a_{n-1}x^{n-1}+\dots+a_1x+a_0$ be a polynomial in $\mathbb{Z}[x]$ . If there exists a prime $p$ such that:

(i) $p\mid a_i$ for $i\neq n$ ,
(ii) $p\nmid a_n$ , and
(iii) $p^2\nmid a_0$

then $f$ is irreducible over $\mathbb{Q}$ .

One way to remember Eisenstein’s Criterion is to remember this classic application to show the irreducibility of the cyclotomic polynomials (after substituting $x+1$ for $x$ ):

$\displaystyle\frac{(x+1)^p-1}{x}=x^{p-1}+{p\choose{p-1}}x^{p-2}+\dots+{p\choose 2}x+{p\choose 1}$ .

Author: mathtuition88