Cohn's irreducibility criterion

Arthur Cohn's irreducibility criterion is a test to determine whether a polynomial is irreducible in $\mathbb{Z}[x]$.

The criterion is often stated as follows:

If a prime number p is expressed in base 10 as $p=a_m10^m+a_{m-1}10^{m-1}+\dots+a_110+a_0$ (where $0\leq a_i\leq 9$) then the polynomial

$f(x)=a_mx^m+a_{m-1}x^{m-1}+\dots+a_1x+a_0$

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 $p(x)=a_kx^k+a_{k-1}x^{k-1}+\dots+a_1x+a_0$ 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. ^ George Pólya; Gábor Szegő (1925). Aufgaben und Lehrsätze aus der Analysis, Bd 2. Springer, Berlin. OCLC 73165700.  English translation in: George Pólya; Gabor Szegö (2004). Problems and theorems in analysis, volume 2. 2. Springer. p. 137. ISBN 3-540-63686-2. 
2. ^ Brillhart, John; Michael Filaseta, Andrew Odlyzko (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. http://www.mast.queensu.ca/~murty/polya4.dvi.  (dvi file)
4. ^ Arthur Cohn's entry at the Mathematics Genealogy Project

External links

• A. Cohn's irreducibility criterion on PlanetMath