Cohn's irreducibility criterion


Arthur Cohn's irreducibility criterion is a sufficient condition for a polynomial to be irreducible in polynomial ring|—that is, for it to be unfactorable into the product of lower-degree polynomials with integer coefficients.
The criterion is often stated as follows:
The theorem can be generalized to other bases as follows:
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