Cohn's irreducibility criterion

Cohn's irreducibility criterion is a sufficient condition for a polynomial to be irreducible in Z [ x ] {\displaystyle \mathbb {Z} [x]} —that is, for it to be unfactorable into the product of lower-degree polynomials with integer coefficients. == Statement == The criterion is often stated as follows: If a prime number p {\displaystyle p} is expressed in base 10 as p = a m 10 m + a m − 1 10 m − 1 + ⋯ + a 1 10 + a 0 {\displaystyle p=a_{m}10^{m}+a_{m-1}10^{m-1}+\cdots +a_{1}10+a_{0}} (where 0 ≤ a i ≤ 9 {\displaystyle 0\leq a_{i}\leq 9} ) then the polynomial f ( x ) = a m x m + a m − 1 x m − 1 + ⋯ + a 1 x + a 0 {\displaystyle f(x)=a_{m}x^{m}+a_{m-1}x^{m-1}+\cdots +a_{1}x+a_{0}} is irreducible in Z [ x ] {\displaystyle \mathbb {Z} [x]} .

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