Perron's irreducibility criterion is a sufficient condition for a
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An ex ...
to be
irreducible in
—that is, for it to be unfactorable into the product of lower-
degree
Degree may refer to:
As a unit of measurement
* Degree (angle), a unit of angle measurement
** Degree of geographical latitude
** Degree of geographical longitude
* Degree symbol (°), a notation used in science, engineering, and mathemati ...
polynomials with
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
coefficient
In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or an expression; it is usually a number, but may be any expression (including variables such as , and ). When the coefficients are themselves ...
s.
This criterion is applicable only to
monic polynomial
In algebra, a monic polynomial is a single-variable polynomial (that is, a univariate polynomial) in which the leading coefficient (the nonzero coefficient of highest degree) is equal to 1. Therefore, a monic polynomial has the form:
:x^n+c_x^+\ ...
s. However, unlike other commonly used criteria, Perron's criterion does not require any knowledge of
prime decomposition
In number theory, integer factorization is the decomposition of a composite number into a product of smaller integers. If these factors are further restricted to prime numbers, the process is called prime factorization.
When the numbers are su ...
of the polynomial's coefficients.
Criterion
Suppose we have the following
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An ex ...
with integer coefficients
:
where
. If either of the following two conditions applies:
*
*
then
is
irreducible over the integers (and by
Gauss's lemma also over the
rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...
s).
History
The criterion was first published by
Oskar Perron
Oskar Perron (7 May 1880 – 22 February 1975) was a German mathematician.
He was a professor at the University of Heidelberg from 1914 to 1922 and at the University of Munich from 1922 to 1951. He made numerous contributions to differen ...
in 1907 in
Journal für die reine und angewandte Mathematik
''Crelle's Journal'', or just ''Crelle'', is the common name for a mathematics journal, the ''Journal für die reine und angewandte Mathematik'' (in English language, English: ''Journal for Pure and Applied Mathematics'').
History
The journal wa ...
.
Proof
A short
proof
Proof most often refers to:
* Proof (truth), argument or sufficient evidence for the truth of a proposition
* Alcohol proof, a measure of an alcoholic drink's strength
Proof may also refer to:
Mathematics and formal logic
* Formal proof, a con ...
can be given based on the following
lemma
Lemma may refer to:
Language and linguistics
* Lemma (morphology), the canonical, dictionary or citation form of a word
* Lemma (psycholinguistics), a mental abstraction of a word about to be uttered
Science and mathematics
* Lemma (botany), a ...
due to Panaitopol:
[. vol. XCVIII no. 10, 39–340]
Lemma. Let
be a polynomial with
. Then exactly one
zero
0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by multiplying digits to the left of 0 by the radix, usu ...
of
satisfies
, and the other
zeroes of
satisfy
.
Suppose that
where
and
are integer polynomials. Since, by the above lemma,
has only one zero with
modulus not less than
, one of the polynomials
has all its zeroes strictly inside the
unit circle
In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucli ...
. Suppose that
are the zeroes of
, and
. Note that
is a nonzero integer, and
, contradiction. Therefore,
is irreducible.
Generalizations
In his publication Perron provided variants of the criterion for multivariate polynomials over arbitrary
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
s. In 2010, Bonciocat published novel proofs of these criteria.
[{{cite book , last1=Bonciocat , first1=Nicolae , title=On an irreducibility criterion of Perron for multivariate polynomials , year=2010 , publisher=Societatea de Științe Matematice din România , oclc=6733580644 ]
See also
*
Eisenstein's criterion
In mathematics, Eisenstein's criterion gives a sufficient condition for a polynomial with integer coefficients to be irreducible over the rational numbers – that is, for it to not be factorizable into the product of non-constant polynomials wit ...
*
Cohn's irreducibility criterion
References
Polynomials
Theorems in algebra