Cohn's irreducibility criterion
Arthur Cohn's irreducibility criterion is a sufficient condition for a polynomial to be irreducible in 
The criterion is often stated as follows:
- If a prime number
is expressed in base 10 as
(where
) then the polynomial
- is irreducible in
The theorem can be generalized to other bases as follows:
- Assume that
is a natural number and
is a polynomial such that
. If
is a prime number then
is irreducible in
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.
A further generalization of the theorem allowing coefficients larger than digits was given by Filaseta and Gross. In particular, let 
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) 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.