# 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 $p=a_m10^m+a_{m-1}10^{m-1}+\cdots+a_110+a_0$ (where $0\leq a_i\leq 9$) then the polynomial
$f(x)=a_mx^m+a_{m-1}x^{m-1}+\cdots+a_1x+a_0$
is irreducible in $\mathbb{Z}[x]$.

The theorem can be generalized to other bases as follows:

Assume that $b \ge 2$ is a natural number and $p(x)=a_kx^k+a_{k-1}x^{k-1}+\cdots+a_1x+a_0$ 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 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, a student of Issai Schur who was awarded his PhD in Berlin in 1921.[4]

## References

1. ^ Pólya, George; Szegő, Gábor (1925). Aufgaben und Lehrsätze aus der Analysis, Bd 2. Springer, Berlin. OCLC 73165700. English translation in: Pólya, George; Szegő, Gábor (2004). Problems and theorems in analysis, volume 2 2. Springer. p. 137. ISBN 3-540-63686-2.
2. ^ Brillhart, John; Filaseta, Michael; Odlyzko, Andrew (1981). "On an irreducibility theorem of A. Cohn". Canadian Journal of Mathematics 33 (5): 1055–1059. doi:10.4153/CJM-1981-080-0.
3. ^ Murty, Ram (2002). "Prime Numbers and Irreducible Polynomials". American Mathematical Monthly (The American Mathematical Monthly, Vol. 109, No. 5) 109 (5): 452–458. doi:10.2307/2695645. JSTOR 2695645. (dvi file)
4. ^ Arthur Cohn's entry at the Mathematics Genealogy Project