Cohn's irreducibility criterion

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Arthur Cohn's irreducibility criterion is a test to determine whether a polynomial is irreducible in the integers.

In base 10, the criterion can be stated as follows:

p

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 criterion is often stated using base 10, as above, but in fact it holds in any base. The general statement of the criterion in base b is:

Assume that $b \ge 2$ is an integer 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 version of the theorem attributed to Cohn by Pólya and Szegö (1925) is only for base 10. ^[1] The first generalization to any base 2 or greater (without side conditions) is due to Brillhart, Filaseta and Odlyzko (1981) ^[2]

Ram Murty gave a simplified proof for the base 2 case in 2002, in a paper that is available on-line and includes more of the history.^[3].

[edit] References

^ George Pólya; Gabor Szegö (1925). Aufgaben und Lehrsätze aus der Analysis, volume 2. Springer, Berlin. OCLC 73165700. English translation in: George Pólya; Gabor Szegö (2004). Problems and theorems in analysis, volume 2. Springer. ISBN 3-540-63686-2.
^ Brillhart, John; Michael Filaseta, Andrew Odlyzko. "On an irreducibility theorem of A. Cohn". Canadian Journal of Mathematics Vol. 33 (1981) (no. 5): 1055-1059.
^ Murty, Ram. "Prime Numbers and Irreducible Polynomials". American Mathematical Monthly Vol. 109 (2002) (no. 5): 452-458. (dvi file)