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• Irreducible_polynomial abstract "In mathematics, a polynomial is said to be irreducible if it cannot be 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 coefficients, a polynomial is irreducible if it cannot be expressed as the product of two or more such polynomials, each of them having a lower degree than the original one. For example, while is reducible over the rationals, is not.For any 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. 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 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 arenon-invertible and non zero.It is helpful to compare irreducible polynomials to prime numbers: prime numbers (together with the corresponding negative numbers of equal magnitude) are the irreducible integers. 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: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. 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. 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.".
• Irreducible_polynomial wikiPageExternalLink books?id=xqMqxQTFUkMC&pg=PA91.
• Irreducible_polynomial wikiPageExternalLink books?id=nSzoG72E93MC&pg=PA154.
• Irreducible_polynomial wikiPageExternalLink PolyInfo.html.
• Irreducible_polynomial wikiPageID "188725".
• Irreducible_polynomial wikiPageRevisionID "591531966".
• Irreducible_polynomial hasPhotoCollection Irreducible_polynomial.
• Irreducible_polynomial title "Irreducible Polynomial".
• Irreducible_polynomial urlname "IrreduciblePolynomial".
• Irreducible_polynomial urlname "IrreduciblePolynomial2".
• Irreducible_polynomial subject Category:Abstract_algebra.
• Irreducible_polynomial subject Category:Algebra.
• Irreducible_polynomial subject Category:Polynomials.
• Irreducible_polynomial type Abstraction100002137.
• Irreducible_polynomial type Function113783816.
• Irreducible_polynomial type MathematicalRelation113783581.
• Irreducible_polynomial type Polynomial105861855.
• Irreducible_polynomial type Polynomials.
• Irreducible_polynomial type Relation100031921.
• Irreducible_polynomial comment "In mathematics, a polynomial is said to be irreducible if it cannot be 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 coefficients, a polynomial is irreducible if it cannot be expressed as the product of two or more such polynomials, each of them having a lower degree than the original one.".
• Irreducible_polynomial label "Factorisation des polynômes".
• Irreducible_polynomial label "Irreducible polynomial".
• Irreducible_polynomial label "Irreduzibles Polynom".
• Irreducible_polynomial label "Polinomio irreducible".
• Irreducible_polynomial label "Polinomio irriducibile".
• Irreducible_polynomial label "Polinômio irredutível".
• Irreducible_polynomial label "Wielomian nierozkładalny".
• Irreducible_polynomial label "Неприводимый многочлен".
• Irreducible_polynomial sameAs Ireducibilní_polynom.
• Irreducible_polynomial sameAs Irreduzibles_Polynom.
• Irreducible_polynomial sameAs Polinomio_irreducible.
• Irreducible_polynomial sameAs Factorisation_des_polynômes.
• Irreducible_polynomial sameAs Polinomio_irriducibile.
• Irreducible_polynomial sameAs Wielomian_nierozkładalny.
• Irreducible_polynomial sameAs Polinômio_irredutível.
• Irreducible_polynomial sameAs m.019trd.
• Irreducible_polynomial sameAs Q1476663.
• Irreducible_polynomial sameAs Q1476663.
• Irreducible_polynomial sameAs Irreducible_polynomial.
• Irreducible_polynomial wasDerivedFrom Irreducible_polynomial?oldid=591531966.
• Irreducible_polynomial isPrimaryTopicOf Irreducible_polynomial.