Cohn's irreducibility criterion

From Wikipedia, the free encyclopedia

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].

[edit] 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]

[edit] References

^ 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, 137. ISBN 3-540-63686-2.
^ Brillhart, John; Michael Filaseta, Andrew Odlyzko (1981). "On an irreducibility theorem of A. Cohn". Canadian Journal of Mathematics 33 (5): 1055–1059.
^ Murty, Ram (2002). "Prime Numbers and Irreducible Polynomials". American Mathematical Monthly 109 (5): 452–458. doi:10.2307/2695645. (dvi file)
^ Arthur Cohn's entry at the Mathematics Genealogy Project