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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク discriminant ： ウィキペディア英語版 discriminant In algebra, the discriminant of a polynomial is a function of its coefficients, typically denoted by a capital 'D' or the capital Greek letter Delta (Δ). It gives information about the nature of its roots. Typically, the discriminant is zero if and only if the polynomial has a multiple root. For example, the discriminant of the quadratic polynomial :$ax^2+bx+c\backslash ,$ is :$\backslash Delta\; =\; \backslash ,b^2-4ac.$ Here for real a, b and c, if Δ > 0, the polynomial has two real roots, if Δ = 0, the polynomial has one real double root, and if Δ < 0, the two roots of the polynomial are complex conjugates. The discriminant of the cubic polynomial :$ax^3+bx^2+cx+d\backslash ,$ is :$\backslash Delta\; =\; \backslash ,b^2c^2-4ac^3-4b^3d-27a^2d^2+18abcd.$ For higher degrees, the discriminant is always a polynomial function of the coefficients. It becomes significantly longer for the higher degrees. The discriminant of a ''general'' quartic has 16 terms,〔, (Chapter 10 page 180 ) 〕 that of a quintic has 59 terms,〔, (Preview page 1 ) 〕 that of a 6th degree polynomial has 246 terms,〔, (Chapter 1 page 26 ) 〕 and the number of terms increases exponentially with the degree. A polynomial has a multiple root (i.e. a root with multiplicity greater than one) in the complex numbers if and only if its discriminant is zero. The concept also applies if the polynomial has coefficients in a field which is not contained in the complex numbers. In this case, the discriminant vanishes if and only if the polynomial has a multiple root in any algebraically closed field containing the coefficients. As the discriminant is a polynomial function of the coefficients, it is defined as long as the coefficients belong to an integral domain ''R'' and, in this case, the discriminant is in ''R''. In particular, the discriminant of a polynomial with integer coefficients is always an integer. This property is widely used in number theory. The term "discriminant" was coined in 1851 by the British mathematician James Joseph Sylvester.〔J. J. Sylvester (1851) "On a remarkable discovery in the theory of canonical forms and of hyperdeterminants," ''Philosophical Magazine'', 4th series, 2 : 391-410; Sylvester coins the word "discriminant" on (page 406 ).〕 ==Definition== In terms of the roots, the discriminant is given by :$\backslash Delta\; =\; a\_n^\backslash prod\_=(-1)^a\_n^\backslash prod\_$ where $a\_n$ is the leading coefficient and $r\_1,\; \backslash ldots,\; r\_n$ are the roots (counting multiplicity) of the polynomial in some splitting field. It is the square of the Vandermonde polynomial times $a\_n^$. As the discriminant is a symmetric function in the roots, it can also be expressed in terms of the coefficients of the polynomial, since the coefficients are the elementary symmetric polynomials in the roots; such a formula is given below. Expressing the discriminant in terms of the roots makes its key property clear, namely that it vanishes if and only if there is a repeated root, but does not allow it to be calculated without factoring a polynomial, after which the information it provides is redundant (if one has the roots, one can tell if there are any duplicates). Hence the formula in terms of the coefficients allows the nature of the roots to be determined without factoring the polynomial. 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「discriminant」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.