Algebraic equation

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 ${\displaystyle x=\frac{1+\sqrt{5}}{2}}$ for the positive solution of ${\displaystyle x^2-x-1=0}$. The ancient Egyptians knew how to solve equations of degree 2 in this manner. The Indian mathematician Brahmagupta (597–668 AD) 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 (univariate) algebraic equations over the rationals (that is, with rational coefficients). 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 (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 ${\displaystyle P=Q}$ is equivalent to ${\displaystyle P-Q=0}$. 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 ${\displaystyle y^4+\frac{xy}{2}=\frac{x^3}{3}-x{y}^2+y^2-\frac{1}{7}}$ becomes

${\displaystyle 42y^4+21xy-14x^3+42x{y}^2-42y^2+6=0.}$

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

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

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.

Theory

Polynomials

Given an equation in unknown x

${\displaystyle (\mathrm{E})a_n{x}^n+a_{n-1}{x}^{n-1}+\cdots +a_{1}x+a_0=0}$,

with coefficients in a field K, one can equivalently say that the solutions of (E) in K are the roots in K of the polynomial

${\displaystyle P=a_n{X}^n+a_{n-1}{X}^{n-1}+\cdots +a_{1}X+a_0\in K[X]}$.

It can be shown that a polynomial of degree n in a field has at most n roots. The equation (E) therefore has at most n solutions.

If K' is a field extension of K, one may consider (E) to be an equation with coefficients in K and the solutions of (E) in K are also solutions in K' (the converse does not hold in general). It is always possible to find a field extension of K known as the rupture field of the polynomial P, in which (E) 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 ${\displaystyle x^2+1=0}$ does not have a solution in ${\displaystyle \mathbb{R}}$ (the solutions are the imaginary units i and −i).

While the real solutions of real equations are intuitive (they are the x-coordinates of the points where the curve y = P(x) intersects the x-axis), 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 x is continuous, and it approaches ${\displaystyle -\mathrm{\infty }}$ as x approaches ${\displaystyle -\mathrm{\infty }}$ and ${\displaystyle +\mathrm{\infty }}$ as x approaches ${\displaystyle +\mathrm{\infty }}$. By the intermediate value theorem, it must therefore assume the value zero at some real x, 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 (using only the four arithmetic operations and taking roots) 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 n reduces to factoring the associated polynomial, that is, rewriting (E) in the form

${\displaystyle a_n(x-z_1)\dots (x-z_n)=0}$,

where the solutions are then the ${\displaystyle z_1,\dots ,z_n}$. The problem is then to express the ${\displaystyle z_i}$ in terms of the ${\displaystyle a_i}$.

This approach applies more generally if the coefficients and solutions belong to an integral domain.

General techniques

Factoring

If an equation P(x) = 0 of degree n has a rational root α, the associated polynomial can be factored to give the form P(X) = (X − α)Q(X) (by dividing P(X) by X − α or by writing P(X) − P(α) as a linear combination of terms of the form X^k − α^k, and factoring out X − α. Solving P(x) = 0 thus reduces to solving the degree n − 1 equation Q(x) = 0. See for example the case n = 3.

Elimination of the sub-dominant term

To solve an equation of degree n,

${\displaystyle (\mathrm{E})a_n{x}^n+a_{n-1}{x}^{n-1}+\cdots +a_{1}x+a_0=0}$,

a common preliminary step is to eliminate the degree-n - 1 term: by setting ${\displaystyle x=y-\frac{a_{n-1}}{n{a}_n}}$, equation (E) becomes

${\displaystyle a_n{y}^n+b_{n-2}{y}^{n-2}+\cdots +b_{1}y+b_0=0}$.

Leonhard Euler developed this technique for the case n = 3 but it is also applicable to the case n = 4, for example.

Quadratic equations

To solve a quadratic equation of the form ${\displaystyle a{x}^2+bx+c=0}$ one calculates the discriminant Δ defined by ${\displaystyle \mathrm{\Delta }=b^2-4ac}$.

If the polynomial has real coefficients, it has:

• two distinct real roots if ${\displaystyle \mathrm{\Delta }>0}$ ;
• one real double root if ${\displaystyle \mathrm{\Delta }=0}$ ;
• no real root if ${\displaystyle \mathrm{\Delta }<0}$, but two complex conjugate roots.

Cubic equations

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:

• Tschirnhaus transformation (general method, not guaranteed to succeed);
• Bezout method (general method, not guaranteed to succeed);
• Ferrari method (solutions for degree 4);
• Euler method (solutions for degree 4);
• Lagrange method (solutions for degree 4);
• Descartes method (solutions for degree 2 or 4);

A quartic equation ${\displaystyle a{x}^4+b{x}^3+c{x}^2+dx+e=0}$ with ${\displaystyle a\ne 0}$ may be reduced to a quadratic equation by a change of variable provided it is either biquadratic (b = d = 0) or quasi-palindromic (e = a, d = b).

Some cubic and quartic equations can be solved using trigonometry or hyperbolic functions.

Higher-degree equations

Évariste Galois 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.