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) 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 PhD in Berlin in 1921.^[4][5]

See also

• Eisenstein's criterion

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
5. Siegmund-Schultze, Reinhard (2009). Mathematicians Fleeing from Nazi Germany: Individual Fates and Global Impact. Princeton, N.J.: Princeton University Press. p. 346. ISBN 9781400831401.

External links

• A. Cohn's irreducibility criterion at PlanetMath.org.