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In mathematics, an algebraic equation or polynomial equation is an equation of the form : where ''P'' and ''Q'' are polynomials with coefficients in some field, often the field of the rational numbers. For most authors, an algebraic equation is ''univariate'', which means that it involves only one variable. On the other hand, a polynomial equation may involve several variables, in which case it is called ''multivariate'' and the term ''polynomial equation'' is usually preferred to ''algebraic equation''. For example, : is an algebraic equation with integer coefficients and : is a multivariate polynomial equation over the rationals. Some but not all polynomial equations with rational coefficients have a solution that is an algebraic expression with a finite number of operations involving just those coefficients (that is, can be solved algebraically). This can be done for all such equations of degree one, two, three, or four; but for degree five or more it can only be done for some equations but not for all. A large amount of research has been devoted to compute efficiently accurate approximations of the real or complex solutions of an univariate algebraic equation (see Root-finding algorithm) and of the common solutions of several multivariate polynomial equations (see System of polynomial equations). == History == The study of algebraic equations is probably as old as mathematics: the Babylonian mathematicians, as early as 2000 BC could solve some kinds of quadratic equations (displayed on Old Babylonian clay tablets). Univariate algebraic equations over the rationals (i.e., with rational coefficients) have a very long history. Ancient mathematicians wanted the solutions in the form of radical expressions, like for the positive solution of . The ancient Egyptians knew how to solve equations of degree 2 in this manner. In the 9th century Muhammad ibn Musa al-Khwarizmi and other Islamic mathematicians derived the quadratic formula, the general solution of equations of degree 2, and recognized the importance of the discriminant. During the Renaissance in 1545, Gerolamo Cardano published the solution of Scipione del Ferro and Niccolò Fontana Tartaglia to equations of degree 3 and that of Lodovico Ferrari for equations of degree 4. Finally Niels Henrik Abel proved, in 1824, that equations of degree 5 and higher do not have general solutions using radicals. Galois theory, named after Évariste Galois, showed that some equations of at least degree 5 do not even have an idiosyncratic solution in radicals, and gave criteria for deciding if an equation is in fact solvable using radicals. 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「algebraic equation」の詳細全文を読む スポンサード リンク
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