# Cohn's irreducibility criterion

Arthur Cohn'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 prime number $p$ is expressed in base 10 as $p=a_{m}10^{m}+a_{m-1}10^{m-1}+\cdots +a_{1}10+a_{0}$ (where $0\leq a_{i}\leq 9$ ) then the polynomial
$f(x)=a_{m}x^{m}+a_{m-1}x^{m-1}+\cdots +a_{1}x+a_{0}$ is irreducible in $\mathbb {Z} [x]$ .

The theorem can be generalized to other bases as follows:

Assume that $b\geq 2$ is a natural number and $p(x)=a_{k}x^{k}+a_{k-1}x^{k-1}+\cdots +a_{1}x+a_{0}$ is a polynomial such that $0\leq a_{i}\leq b-1$ . If $p(b)$ is a prime number then $p(x)$ is irreducible in $\mathbb {Z} [x]$ .

The base-10 version of the theorem is attributed to Cohn by Pólya and Szegő in one of their books while the generalization to any base b is due to Brillhart, Filaseta, and Odlyzko.

In 2002, Ram Murty gave a simplified proof as well as some history of the theorem in a paper that is available online.

The converse of this criterion is that, if p is an irreducible polynomial with integer coefficients that have greatest common divisor 1, then there exists a base such that the coefficients of p form the representation of a prime number in that base; this is the Bunyakovsky conjecture and its truth or falsity remains an open question.

## Historical notes

• Polya and Szegő gave their own generalization but it has many side conditions (on the locations of the roots, for instance)[citation needed] so it lacks the elegance of Brillhart's, Filaseta's, and Odlyzko's generalization.
• It is clear from context that the "A. Cohn" mentioned by Polya and Szegő is Arthur Cohn (1894–1940), a student of Issai Schur who was awarded his doctorate from Frederick William University in 1921.