Algebraic equation

In mathematics, an algebraic equation or polynomial equation is an equation of the form

P = Q

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,

x^5-3x+1=0

is an algebraic equation with integer coefficients and

y^4+\frac{xy}{2}=\frac{x^3}{3}-xy^2+y^2-\frac{1}{7}

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 x=\frac{1+\sqrt{5}}{2} for the positive solution of x^2-x-1=0. 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.

Areas of study

The algebraic equations are the basis of a number of areas of modern mathematics: Algebraic number theory is the study of (univariate) algebraic equations over the rationals (that is, with rational coefficients). Galois theory has been introduced by Évariste Galois for getting criteria deciding if an algebraic equation may be solved in terms of radicals. In field theory, an algebraic extension is an extension such that every element is a root of an algebraic equation over the base field. Transcendental number theory is the study of the real numbers which are not solutions to an algebraic equation over the rationals. A Diophantine equation is a (usually multivariate) polynomial equation with integer coefficients for which one is interested in the integer solutions. Algebraic geometry is the study of the solutions in an algebraically closed field of multivariate polynomial equations.

Two equations are equivalent if they have the same set of solutions. In particular the equation P = Q is equivalent with P-Q = 0. It follows that the study of algebraic equations is equivalent to the study of polynomials.

A polynomial equation over the rationals can always be converted to an equivalent one in which the coefficients are integers. For example, multiplying through by 42 = 2·3·7 and grouping its terms in the first member, the previously mentioned polynomial equation y^4+\frac{xy}{2}=\frac{x^3}{3}-xy^2+y^2-\frac{1}{7} becomes

42y^4+21xy-14x^3+42xy^2-42y^2+6=0.

Because sine, exponentiation, and 1/T are not polynomial functions,

e^T x^2+\frac{1}{T}xy+\sin(T)z -2 =0

is not a polynomial equation in the four variables x, y, z, and T over the rational numbers. However, it is a polynomial equation in the three variables x, y, and z over the field of the elementary functions in the variable T.

Solutions

As for any equation, the solutions of an equation are the values of the variables for which the equation is true. For univariate algebraic equations these are also called roots, even if, properly speaking, one should say the solutions of the algebraic equation P=0 are the roots of the polynomial P. When solving an equation, it is important to specify in which set the solutions are allowed. For example, for an equation over the rationals one may look for solutions in which all the variables are integers. In this case the equation is a Diophantine equation. One may also be interested only in the real solutions. However, for univariate algebraic equations, the number of solutions is finite, and all solutions are contained in any algebraically closed field containing the coefficients—for example, the field of complex numbers in the case of equations over the rationals. It follows that without precision "root" and "solution" usually mean "solution in an algebraically closed field".

See also

References

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