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Irreducible polynomial
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{{short description|Polynomial without nontrivial factorization}} {{about|non-factorizable polynomials|polynomials which are not a composition of polynomials|Indecomposable polynomial}} {{more footnotes|date=March 2015}} In [[mathematics]], an '''irreducible polynomial''' is, roughly speaking, a [[polynomial]] that cannot be [[factored]] into the product of two [[Constant polynomial|non-constant polynomials]]. The property of irreducibility depends on the nature of the [[coefficient]]s that are accepted for the possible factors, that is, the [[ring (mathematics)|ring]] to which the [[coefficient]]s of the polynomial and its possible factors are supposed to belong. For example, the polynomial {{math|''x''<sup>2</sup> β 2}} is a polynomial with [[integer]] coefficients, but, as every integer is also a [[real number]], it is also a polynomial with real coefficients. It is irreducible if it is considered as a polynomial with integer coefficients, but it factors as <math>\left(x - \sqrt{2}\right)\left(x + \sqrt{2}\right)</math> if it is considered as a polynomial with real coefficients. One says that the polynomial {{math|''x''<sup>2</sup> β 2}} is irreducible over the integers but not over the reals. Polynomial irreducibility can be considered for polynomials with coefficients in an [[integral domain]], and there are two common definitions. Most often, a polynomial over an integral domain {{mvar|R}} is said to be ''irreducible'' if it is not the product of two polynomials that have their coefficients in {{mvar|R}}, and that are not [[unit (ring theory)|unit]] in {{mvar|R}}. Equivalently, for this definition, an irreducible polynomial is an [[irreducible element]] in a ring of polynomials over {{mvar|R}}. If {{mvar|R}} is a field, the two definitions of irreducibility are equivalent. For the second definition, a polynomial is irreducible if it cannot be factored into polynomials with coefficients in the same domain that both have a positive degree. Equivalently, a polynomial is irreducible if it is irreducible over the [[field of fractions]] of the integral domain. For example, the polynomial <math>2(x^2-2)\in \Z[x]</math> is irreducible for the second definition, and not for the first one. On the other hand, <math>x^2-2</math> is irreducible in <math>\Z[x]</math> for the two definitions, while it is reducible in <math>\R[x].</math> A polynomial that is irreducible over any field containing the coefficients is [[absolutely irreducible]]. By the [[fundamental theorem of algebra]], a [[univariate polynomial]] is absolutely irreducible if and only if its degree is one. On the other hand, with several [[indeterminate (variable)|indeterminate]]s, there are absolutely irreducible polynomials of any degree, such as <math>x^2 + y^n - 1,</math> for any positive integer {{math|''n''}}. A polynomial that is not irreducible is sometimes said to be a '''reducible polynomial'''.<ref>{{harvnb|Gallian|2012|p=311}}</ref><ref>{{harvnb|Mac Lane|Birkhoff|1999}} do not explicitly define "reducible", but they use it in several places. For example: "For the present, we note only that any reducible quadratic or cubic polynomial must have a linear factor." (p. 268).</ref> Irreducible polynomials appear naturally in the study of [[polynomial factorization]] and [[algebraic field extension]]s. It is helpful to compare irreducible polynomials to [[prime number]]s: prime numbers (together with the corresponding negative numbers of equal magnitude) are the irreducible [[integer]]s. They exhibit many of the general properties of the concept of "irreducibility" that equally apply to irreducible polynomials, such as the essentially unique factorization into prime or irreducible factors. When the coefficient ring is a field or other [[unique factorization domain]], an irreducible polynomial is also called a '''prime polynomial''', because it generates a [[prime ideal]].
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