# Cohn's irreducibility criterion

Arthur Cohn's irreducibility criterion is a sufficient condition for a polynomial to be irreducible in ${\displaystyle \mathbb {Z} [x]}$—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 ${\displaystyle p}$ is expressed in base 10 as ${\displaystyle p=a_{m}10^{m}+a_{m-1}10^{m-1}+\cdots +a_{1}10+a_{0}}$ (where ${\displaystyle 0\leq a_{i}\leq 9}$) then the polynomial
${\displaystyle f(x)=a_{m}x^{m}+a_{m-1}x^{m-1}+\cdots +a_{1}x+a_{0}}$
is irreducible in ${\displaystyle \mathbb {Z} [x]}$.

The theorem can be generalized to other bases as follows:

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

The base-10 version of the theorem is attributed to Cohn by Pólya and Szegő in one of their books[1] while the generalization to any base, 2 or greater, is due to Brillhart, Filaseta, and Odlyzko.[2]

In 2002, Ram Murty gave a simplified proof as well as some history of the theorem in a paper that is available online.[3]

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.[4][5]