User:SilTheFirst/sandbox

Oskar Perron's irreducibility criterion is a sufficient condition for a polynomial to be irreducible in $\mathbb{Z}[x]$—that is, for it to be unfactorable into the product of lower-degree polynomials with integer coefficients.

The criterion is often stated as follows:
 * If a polynomial $$f(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$$ with integer coefficients satisfies $$|a_{n-1}|> 1+|a_{n-2}|+\cdots+|a_0|$$ and $$a_0 \neq 0$$, then it is irreducible in $$\mathbb{Z}[x]$$.

Sometimes the following variant of the theorem is used:
 * If a polynomial $$f(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$$ with integer coefficients satisfies $$|a_{n-1}|\geq 1+|a_{n-2}|+\cdots+|a_0|$$, $$f(\pm 1) \neq 0$$ and $$a_0 \neq 0$$, then it is irreducible in $$\mathbb{Z}[x]$$.

Unlike other commonly used criteria, Perron's criterion does not require any knowledge of prime factorization of the polynomial's coefficients.

Historical notes

 * First appeareance of the criterion is attributed to Perron and his publication in 1907.


 * In 2010, variant for multivariate polynomials was published by Bonciocat.