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In mathematics, an irreducible polynomial is, roughly speaking, a non-constant polynomial that cannot be factored into the product of two non-constant polynomials. The property of irreducibility depends on the field or ring to which the coefficients are considered to belong. For example, the polynomial is irreducible if the coefficients 1 and -2 are considered as integers and factors as if the coefficients are considered as real numbers. One says "the polynomial is irreducible over the integers but not over the reals". A polynomial that is not irreducible is sometimes said to be reducible.〔Gallian 2012, p. 311〕〔Mac Lane and 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)〕 However this term must be used with care, as it may refer to other notions of reduction. Irreducible polynomials appear naturally in polynomial factorization and algebraic field extensions. 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. ==Definition== If ''F'' is a field, a non-constant polynomial is irreducible over ''F'' if its coefficients belong to ''F'' and it cannot be factored into the product of two non-constant polynomials with coefficients in ''F''. A polynomial with integer coefficients, or, more generally, with coefficients in a unique factorization domain ''R'' is sometimes said to be ''irreducible over R'' if it is an irreducible element of the polynomial ring (a polynomial ring over a unique factorization domain is also a unique factorization domain), that is, it is not invertible, nor zero and cannot be factored into the product of two non-invertible polynomials with coefficients in ''R''. Another definition is frequently used, saying that a polynomial is ''irreducible over R'' if it is irreducible over the field of fractions of ''R'' (the field of rational numbers, if ''R'' is the integers). Both definitions generalize 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 non-invertible and non zero. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Irreducible polynomial」の詳細全文を読む スポンサード リンク
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