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| In [[mathematics]], a [[polynomial]] is said to be '''irreducible''' if it cannot be [[factorization|factored]] into the product of two or more non-trivial polynomials whose coefficients are of a specified type. Thus in the common context of polynomials with [[rational number|rational]] [[coefficient]]s, a polynomial is irreducible if it cannot be expressed as the product of two or more such polynomials, each of them having a lower [[polynomial degree|degree]] than the original one. For example, while <math>x^2-1 = (x-1)(x+1)</math> is reducible over the rationals, <math>x^2+1</math> is not.
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| For any [[field (mathematics)|field]] ''F'', a polynomial with coefficients in ''F'' is said to be '''irreducible''' over ''F'' if it is non-constant and cannot be factored into the product of two or more non-constant polynomials with coefficients in ''F''. The property of irreducibility depends on the field ''F''; a polynomial may be irreducible over some fields but reducible over others. Some simple examples are discussed below.
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| A polynomial with integer coefficients, or, more generally, with coefficients in a [[unique factorization domain]] ''F'' is said to be '''irreducible''' over ''F'' if it is not [[unit (ring theory)|invertible]] nor zero and cannot be factored into the product of two non-invertible polynomials with coefficients in ''F''. This definition generalizes the definition given for the case of coefficients in a field, because, in this case, the non constant polynomials are exactly the polynomials that are
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| non-invertible and non zero.
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| 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:
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| Every polynomial with coefficients in a field or a unique factorization domain ''F'' can be factorized into polynomials that are irreducible over ''F''. This factorization is unique [[up to]] [[permutation]] of the factors and the multiplication of the factors by invertible constants from ''F''. This property of unique factorization is commonly expressed by saying that the polynomial rings over a field or a unique factorization domain are unique factorization domains. However the existence of such a factorization does not mean that, given a polynomial, the factorization may always be computed: there are fields such that it can not exist any [[algorithm]] to factorize polynomials over these fields.<ref>{{citation |author1=Fröhlich, A.|author2=Shepherson, J. C.|title = On the factorisation of polynomials in a finite number of steps|journal = Mathematische Zeitschrift|volume = 62|issue=1|year = 1955|issn = 0025-5874|doi=10.1007/BF01180640}}
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| </ref> There exist factorization algorithms for the polynomials with coefficients in the rational numbers, in a [[finite field]] or a [[finitely generated field extension]] of these fields. They are described in the article [[Polynomial factorization]].
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| If an [[univariate]] polynomial ''p'' has a root (in some [[field extension]]) which is also a root of an irreducible polynomial ''q'', then ''p'' is a multiple of ''q'', and thus all roots of ''q'' are roots of ''p''; this is [[Abel's irreducibility theorem]]. This implies that the roots of an irreducible polynomial may not be distinguished through algebraic relations. This result is one of the starting points of [[Galois theory]], which has been introduced by [[Évariste Galois]] to study the relationship between the roots of a polynomial.
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| == Simple examples ==
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| The following six polynomials demonstrate some elementary properties of reducible and irreducible polynomials:
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| :<math>p_1(x)=x^2+4x+4\,={(x+2)(x+2)}</math>,
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| :<math>p_2(x)=x^2-4\,={(x-2)(x+2)}</math>,
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| :<math>p_3(x)=9x^2-3\,=3(3x^2-1)\,=3(x\sqrt{3}-1)(x\sqrt{3}+1)</math>,
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| :<math>p_4(x)=x^2-4/9\,=(x-2/3)(x+2/3)</math>,
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| :<math>p_5(x)=x^2-2\,=(x-\sqrt{2})(x+\sqrt{2})</math>,
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| :<math>p_6(x)=x^2+1\,={(x-i)(x+i)}</math>.
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| Over the ring <math>\mathbb Z</math> of [[integer]]s, the first three polynomials are reducible (the third one is reducible because the factor 3 is not invertible in the integers), the last two are irreducible. (The fourth, of course, is not a polynomial over the integers.)
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| Over the field <math>\mathbb Q</math> of [[rational number]]s, the first two and the fourth polynomials are reducible, but the other three polynomials are irreducible (as a polynomial over the rationals, 3 is a [[unit (ring theory)|unit]], and, therefore, does not count as a factor).
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| Over the field <math>\mathbb R</math> of [[real number]]s, the first five polynomials are reducible, but <math>p_6(x)</math> is still irreducible.
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| Over the field <math>\mathbb C</math> of [[complex number]]s, all six polynomials are reducible. In fact, every nonzero polynomial <math>p(x)</math> over <math>\mathbb C</math> can be factored as
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| :<math> p(x) = a(x-z_1)\cdots (x-z_n)</math>
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| where <math>n</math> is the degree, <math>a</math> the leading coefficient and <math>z_1,\dots,z_n</math> the zeros of <math>p(x)</math>. Thus, the only non-constant irreducible polynomials over <math>\mathbb C</math> are [[linear function|linear polynomials]]. This is the [[Fundamental theorem of algebra]].
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| The existence of irreducible polynomials of degree greater than one (without zeros in the original field) historically motivated the [[field extension|extension]] of that original number field so that even these polynomials can be reduced into linear factors: from [[rational number]]s ( <math>\mathbb{Q}</math> ), to the [[real number|real]] subset of the [[algebraic number]]s ( <math>\mathcal{A}\cap\mathbb{R}</math> ), and finally to the algebraic subset of the [[complex number]]s ( <math>\mathcal{A}\cap\mathbb{C}</math> ). After the invention of [[calculus]] those latter two subsets were later extended to all [[real number]]s ( <math>\mathbb{R}</math> ) and all [[complex number]]s ( <math>\mathbb{C}</math> ).
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| For algebraic purposes, the extension from rational numbers to real numbers is too "radical": it introduces [[transcendental number]]s, which 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 [[Mathematical analysis|analysis]]). The set of [[algebraic numbers]] ( <math>\mathcal{A}</math> ) is the [[algebraic closure]] of the rationals, and contains the roots of all polynomials (including ''i'' for instance). This is a [[countable]] field and is strictly contained in the complex numbers – the difference being that this field ( <math>\mathcal{A}</math> ) is "algebraically complete" (as are the [[complex number]]s, <math>\mathbb{C}</math> ) but not analytically complete since it lacks the aforementioned transcendentals.
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| The above paragraph generalizes in that there is a purely algebraic process to [[field extension|extend]] a given field ''F'' with a given polynomial <math>p(x)</math> to a larger field where this polynomial <math>p(x)</math> can be reduced into linear factors. The study of such extensions is the starting point of [[Galois theory]].
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| === Real and complex numbers ===
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| As shown in the examples above, only [[linear|linear polynomial]]s are irreducible over the field of complex numbers (this is a consequence of the [[fundamental theorem of algebra]]). Since [[Complex conjugate root theorem|the complex roots of a real polynomial are in conjugate pairs]], the irreducible polynomials over the field of real numbers are the linear polynomials and the quadratic polynomials with no real roots. For example,
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| <math>x^4 + 1</math> factors over the real numbers as <math>(x^2 + \sqrt{2}x + 1)(x^2 - \sqrt{2}x + 1).</math>
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| === Generalization ===
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| If ''R'' is an [[integral domain]], an element ''f'' of ''R'' which is neither zero nor a unit is called [[irreducible element|irreducible]] if there are no non-units ''g'' and ''h'' with ''f'' = ''gh''. One can show that every [[prime element]] is irreducible;<ref>Consider p a prime that is reducible: p=ab. Then p | ab => p | a or p | b. Say p | a => a = pc, then we have: p=ab=pcb => p(1-cb)=0. Because R is a domain we have: cb=1. So b is a unit and p is irreducible</ref> the converse is not true in general but holds in [[unique factorization domain]]s. The [[polynomial ring]] ''F''[''x''] over a field ''F'' (or any unique-factorization domain) is again a unique factorization domain. Inductively, this means that the polynomial ring in ''n'' indeterminants (over a ring ''R'') is a unique factorization domain if the same is true for ''R''.
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| === Finite fields ===
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| Factorization over a [[finite field]] behaves similarly to factorization over the rational or the complex field. However, polynomials with integer coefficients that are irreducible over the field <math>\mathbb Q</math> can be reducible over a finite field. For example, the polynomial <math>x^2+1</math> is irreducible over <math>\mathbb Q</math> but reducible over the field <math>\mathbb F_2</math> of two elements. Indeed, over <math>\mathbb F_2</math>, we have
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| :<math> (x^2+1) = (x+1)^2 </math> | |
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| The irreducibility of a polynomial over the integers <math>\mathbb Z</math> is related to that over the field <math>\mathbb F_p</math> of <math>p</math> elements (for a prime <math>p</math>). Namely, if a polynomial over <math>\mathbb Z</math> with leading coefficient <math>1</math> is reducible over <math>\mathbb Z</math> then it is reducible over <math>\mathbb F_p</math> for any prime <math>p</math>. The converse, however, is not true,<ref>{{cite book|title=Abtract Algebra|year=2004|publisher=John Wiley & Sons, Inc.|isbn=0-471-43334-9|page=309|author=David Dummit|coauthors=Richard Foote|chapter=chapter 9, Proposition 12}}</ref> there are polynomials of arbitrary large degree that are irreducible over the integers and reducible over every finite field. A simple example of such a polynomial is <math>x^4+1,</math> which is irreducible over the integers and reducible over every finite field.
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| == See also ==
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| * [[Gauss's lemma (polynomial)]]
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| * [[Rational root theorem]], a method of finding whether a polynomial has a linear factor with rational coefficients
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| * [[Eisenstein's criterion]]
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| * [[Perron method]]
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| * [[Hilbert's irreducibility theorem]]
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| * [[Cohn's irreducibility criterion]]
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| * [[Irreducible component]] of a [[topological space]]
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| * [[Factorization of polynomials over finite fields]]
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| * [[Quartic function#Factorization into quadratics]]
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| * [[Cubic function#Factorization]]
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| * [[Casus irreducibilis]], the irreducible cubic with three real roots
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| * [[Quadratic equation#Quadratic factorization]]
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| == References ==
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| * {{citation | first1 = Alfred J. | last1 = Menezes | authorlink1 = Alfred Menezes | first2 = Paul C. | last2 = Van Oorschot | authorlink2 = Paul van Oorschot | first3 = Scott A. | last3 = Vanstone | authorlink3 = Scott Vanstone | title = Handbook of applied cryptography | publisher = [[CRC Press]] | year = 1997 | isbn = 978-0-8493-8523-0}}, [http://books.google.com/books?id=nSzoG72E93MC&pg=PA154 pp. 154].
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| * {{citation | first1 = Rudolf | last1 = Lidl | first2 = Harald | last2 = Niederreiter | title = Finite fields | edition = 2nd | publisher = [[Cambridge University Press]] | year = 1997 | isbn = 978-0-521-39231-0}}, [http://books.google.ca/books?id=xqMqxQTFUkMC&pg=PA91 pp. 91].
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| == External links ==
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| * {{MathWorld | title = Irreducible Polynomial | urlname = IrreduciblePolynomial}}
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| * {{PlanetMath | urlname = IrreduciblePolynomial2 | title = Irreducible Polynomial}}
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| * [http://theory.cs.uvic.ca/inf/neck/PolyInfo.html Information on Primitive and Irreducible Polynomials], The (Combinatorial) Object Server.
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| == Notes ==
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| {{reflist}}
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| {{DEFAULTSORT:Irreducible Polynomial}}
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| [[Category:Polynomials]]
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| [[Category:Abstract algebra]]
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| [[Category:Algebra]]
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Alyson is what my spouse enjoys to call me but I don't like when individuals use my full title. For many years she's been working as a travel agent. My husband doesn't like it the way I do but what I truly like doing is caving but I don't have the time lately. I've usually cherished living in Kentucky but now I'm considering other options.
Feel free to visit my web page: psychics online (speaking of)