NationMaster - Encyclopedia: Irreducible polynomial

In mathematics, the adjective irreducible means that an object cannot be expressed as a product of at least two non-trivial factors in a given ring. Mathematics is often defined as the study of topics such as quantity, structure, space, and change. ...

For any field F, the ring of polynomials with coefficients in F is denoted by $F [x]$ . A polynomial $p (x)$ in $F [x]$ is called irreducible over $F$ , if it cannot be represented as the product of two or more non-constant polynomials from $F [x]$ .

This definition depends on the field F. Some simple examples will be discussed below.

Galois theory studies the relationship between a field, it's Galois group and its irreducible polynomials in depths. Interesting and non-trivial applications can be found in the study of Finite fields. In mathematics, Galois theory is a branch of abstract algebra. ... In mathematics, a Galois group is a group associated with a certain type of field extension. ... In abstract algebra, a finite field or Galois field (so named in honor of Evariste Galois) is a field that contains only finitely many elements. ...

It is helpful to compare irreducible polynomials to prime numbers: Prime numbers (together with the corresponding negative numbers of equal modulus and 1) are the irreducible integers. They exhibit many of the general properties of the concept 'irreducibility' that equally apply to irreducible polynomials, such as the essentially unique factorization into prime or irreducible factors: In mathematics, a prime number (or prime) is a natural number greater than one whose only positive divisors are one and itself. ...

Every polynomial $p (x)$ in $F [x]$ can be factorized into polynomials that are irreducible over F. Two such factorizations differ only in the order of the factors and in constant factors from F.

Simple examples

The following three polynomials demonstrate some elementary properties of reducible and irreducible polynomials:

$p_1(x)=x^2-4,=(x-2)(x+2)$ ,

$p_2(x)=x^2-2,=(x-sqrt{2})(x+sqrt{2})$ ,

$p_3(x)=x^2+1,=(x-i)(x+i)$ .

Over the field $mathbb{Q}$ of rational numbers, the first polynomial $p 1 (x)$ is reducible, but the other two polynomials are irreducible.

Over the field $mathbb{R}$ of real numbers, the two polynomials $p 1 (x)$ and $p 2 (x)$ are reducible, but $p 3 (x)$ is still irreducible.

Over the field $mathbb{C}$ of complex numbers, all three polynomials are reducible.

In fact over $mathbb{C}$ , every polynomial can be factored into linear factors

$p(z)=a_n (z-z_1)(z-z_2)cdots(z-z_n)$

where $a n$ is the leading coefficient of the polynomial and $z_1,ldots,z_n$ are the zeros of $p (z)$ . Hence, all non-constant irreducible polynomials are of degree 1. This is the Fundamental theorem of algebra. In mathematics, the fundamental theorem of algebra states that every complex polynomial of degree has exactly roots (zeros), counted with multiplicity. ...

Note: The existence of a essentially unique factorization $p 3 (x) = x 2 + 1 = (x - i)(x + i)$ of $p 3 (x)$ into factors that do not belong to $Q [x]$ implies that this polynomial is irreducible over $mathbb{Q}$ : there cannot be another factorization.

These examples demonstrate the relationship between the the zeros of a polynomial (solutions of an algebraic equation) and the factorization of the polynomial into linear factors.

The existence of irreducible polynomials of degree greater than one (without zeros in the original field) historically motivated the extension of that original number field so that even these polynomials can be reduced into linear factors: From rational numbers to real numbers and further to complex numbers.

For algebraic purposes, this extension process is often too 'radical': It introduces trancendent numbers (that are not the solutions of algebraic equations with rational coefficients). These numbers are not needed for the algebraic purpose of factorizing polynomials (but they are necessary for the use of real numbers in Analysis). Thus, there is a purely algebraic process to extend a given field F with a given, over F irreducible, polynomial $p (x)$ to a larger field where this polynomial $p (x)$ can be reduced into linear factors. The study of such extensions is the starting point of Galois theory. An analysis is a critical evaluation, usually made by breaking a subject (either material or intellectual) down into its constituent parts, then describing the parts and their relationship to the whole. ... In mathematics, Galois theory is a branch of abstract algebra. ...

mito

Results from FactBites:

Math Forum - Ask Dr. Math (500 words)
	In answer to your question, that does not mean that the polynomial is irreducible.
	That's because the 3rd degree ones divide x^7 + 1, and 7 is not a divisor of 51; and the 4th degree ones divide either x^15 + 1 or x^5 + 1, and neither of these is a divisor of 51.
	Irreducible polynomials of period 17 or 51 have degree 8, because 17 and 51 are factors of 2^8 - 1 = 255, and no lower power of 2 will do.

PlanetMath: proof that the cyclotomic polynomial is irreducible (258 words)
	, since it splits this polynomial and is generated as an algebra by a single root of the polynomial.
	"proof that the cyclotomic polynomial is irreducible" is owned by djao.
	This is version 6 of proof that the cyclotomic polynomial is irreducible, born on 2002-05-08, modified 2005-04-03.

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