Perron's irreducibility criterion explained

Perron's irreducibility criterion is a sufficient condition for a polynomial to be irreducible in

—that is, for it to be unfactorable into the product of lower-degree polynomials with integer coefficients.

This criterion is applicable only to monic polynomials. However, unlike other commonly used criteria, Perron's criterion does not require any knowledge of prime decomposition of the polynomial's coefficients.

Criterion

Suppose we have the following polynomial with integer coefficients

	n+a
f(x)=x
	n-1

x^n-1+ … +a_1x+a_0,

where

a_{0 ≠}0

. If either of the following two conditions applies:

|a_n-1|>1+|a_n-2|+ … +|a_0|

|a_n-1|=1+|a_n-2|+ … +|a_0|, f(\pm1) ≠ 0

then

is irreducible over the integers (and by Gauss's lemma also over the rational numbers).

History

The criterion was first published by Oskar Perron in 1907 in Journal für die reine und angewandte Mathematik.^[1]

Proof

A short proof can be given based on the following lemma due to Panaitopol:^[2] ^[3]

Lemma. Let

	n+a
f(x)=x
	n-1

x^n-1+ … +a_1x+a₀

be a polynomial with

|a_n-1|>1+|a_n-2|+ … +|a₁|+|a_0|

. Then exactly one zero

satisfies

|z|>1

, and the other

n-1

zeroes of

satisfy

|z|<1

Suppose that

f(x)=g(x)h(x)

where

and

are integer polynomials. Since, by the above lemma,

has only one zero with modulus not less than

, one of the polynomials

g,h

has all its zeroes strictly inside the unit circle. Suppose that

z_1,...,z_k

are the zeroes of

, and

|z_1|,...,|z_k|<1

. Note that

g(0)

is a nonzero integer, and

|g(0)|=|z_{1 …}z_k|<1

, contradiction. Therefore,

is irreducible.

Generalizations

In his publication Perron provided variants of the criterion for multivariate polynomials over arbitrary fields. In 2010, Bonciocat published novel proofs of these criteria.^[4]

References

Perron . Oskar . Oskar Perron . Neue kriterien für die irreduzibilität algebraischer gleichungen . 1907 . Walter de Gruyter . Journal für die reine und angewandte Mathematik . 132 . 288–307.
Book: Panaitopol . Laurențiu . Criteriul lui Perron de ireductibilitate a polinoamelor cu coeficienti intregi . 1993 . Gazeta Matematică . 6733580644 . . vol. XCVIII no. 10, 39–340
Web site: Integer Polynomials . Zhao . Yufei . 2007 .
Book: Bonciocat . Nicolae . On an irreducibility criterion of Perron for multivariate polynomials . 2010 . Societatea de Științe Matematice din România . 6733580644 .