Algebraic equation


In mathematics, an algebraic equation or polynomial equation is an equation of the form
where P is a polynomial 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 that can be found using a finite number of operations that involve only those same types of coefficients. 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, not for all. A large amount of research has been devoted to compute efficiently accurate approximations of the real or complex solutions of a univariate algebraic equation and of the common solutions of several multivariate 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.
Univariate algebraic equations over the rationals 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. The Indian mathematician Brahmagupta explicitly described the quadratic formula in his treatise Brāhmasphuṭasiddhānta published in 628 AD, but written in words instead of symbols. 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 algebraic equations over the rationals. Galois theory was introduced by Évariste Galois to specify criteria for 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 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 is equivalent to. 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 becomes
Because sine, exponentiation, and 1/T are not polynomial functions,
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.

Theory

Polynomials

Given an equation in unknown
with coefficients in a field, one can equivalently say that the solutions of in are the roots in of the polynomial
It can be shown that a polynomial of degree in a field has at most roots. The equation therefore has at most solutions.
If is a field extension of, one may consider to be an equation with coefficients in and the solutions of in are also solutions in . It is always possible to find a field extension of known as the rupture field of the polynomial, in which has at least one solution.

Existence of solutions to real and complex equations

The fundamental theorem of algebra states that the field of the complex numbers is closed algebraically, that is, all polynomial equations with complex coefficients and degree at least one have a solution.
It follows that all polynomial equations of degree 1 or more with real coefficients have a complex solution. On the other hand, an equation such as does not have a solution in .
While the real solutions of real equations are intuitive, the existence of complex solutions to real equations can be surprising and less easy to visualize.
However, a monic polynomial of odd degree must necessarily have a real root. The associated polynomial function in is continuous, and it approaches as approaches and as approaches. By the intermediate value theorem, it must therefore assume the value zero at some real, which is then a solution of the polynomial equation.

Connection to Galois theory

There exist formulas giving the solutions of real or complex polynomials of degree less than or equal to four as a function of their coefficients. Abel showed that it is not possible to find such a formula in general for equations of degree five or higher. Galois theory provides a criterion which allows one to determine whether the solution to a given polynomial equation can be expressed using radicals.

Explicit solution of numerical equations

Approach

The explicit solution of a real or complex equation of degree 1 is trivial. Solving an equation of higher degree reduces to factoring the associated polynomial, that is, rewriting in the form
where the solutions are then the. The problem is then to express the in terms of the .
This approach applies more generally if the coefficients and solutions belong to an integral domain.

General techniques

Factoring

If an equation of degree has a rational root, the associated polynomial can be factored to give the form (by dividing by or by writing as a linear combination of terms of the form, and factoring out. Solving thus reduces to solving the degree equation. See for example the case.

Elimination of the sub-dominant term

To solve an equation of degree,
a common preliminary step is to eliminate the degree- term: by setting, equation becomes
Leonhard Euler developed this technique for the case but it is also applicable to the case, for example.

Quadratic equations

To solve a quadratic equation of the form one calculates the discriminant Δ defined by.
If the polynomial has real coefficients, it has:
The best-known method for solving cubic equations, by writing roots in terms of radicals, is Cardano's formula.

Quartic equations

For detailed discussions of some solution methods see:
A quartic equation with may be reduced to a quadratic equation by a change of variable provided it is either biquadratic or quasi-palindromic.
Some cubic and quartic equations can be solved using trigonometry or hyperbolic functions.

Higher-degree equations

and Niels Henrik Abel showed independently that in general a polynomial of degree 5 or higher is not solvable using radicals. Some particular equations do have solutions, such as those associated with the cyclotomic polynomials of degrees 5 and 17.
Charles Hermite, on the other hand, showed that polynomials of degree 5 are solvable using elliptical functions.
Otherwise, one may find numerical approximations to the roots using root-finding algorithms, such as Newton's method.