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}.

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