Overview

An algebraic equation is an equality between two polynomial expressions in one or more variables. It is usually written as P = Q, where P and Q are polynomials whose coefficients lie in a chosen domain such as a field or ring. By moving all terms to one side the same relation is written P − Q = 0, and solving the equation is equivalent to finding roots of the polynomial P − Q. Which values count as solutions depends on the allowed domain: integers, rationals, real numbers or complex numbers all give different solution sets. The subject lies at the intersection of elementary mathematics and the theory of algebraic structures such as a field, because what manipulations and conclusions are valid depends on the arithmetic available in the coefficient domain.

Formal definition and basic properties

Formally, an algebraic equation over a given coefficient set is an assertion that two polynomials are equal as functions or formal expressions. When studying solution sets one typically considers the polynomial f = P − Q and asks for values of the variables for which f vanishes. A univariate algebraic equation involves a single variable and reduces to the study of a single polynomial; multivariate equations involve two or more variables and naturally lead to systems of polynomial equations. Coefficients of the polynomials are often discussed as coefficients and may be drawn from standard sets such as the integers, rationals, reals or complexes. In many contexts one prefers to work with integer coefficients because denominators can be cleared: an equation with rational coefficients can be multiplied by a common denominator to obtain an equivalent equation whose coefficients are integers; this observation is useful in arithmetic and number-theoretic contexts and connects polynomial equations to the theory of integers.

Degree, multiplicity and the fundamental theorem for univariate polynomials

The degree of an algebraic equation is the highest total degree of any term after the expression is arranged as a single polynomial equal to zero. The degree controls many qualitative properties: for univariate polynomials over the complex numbers the fundamental theorem of algebra guarantees that a nonconstant polynomial of degree n has exactly n complex roots when counted with multiplicity. A root of multiplicity greater than one is called a repeated root; repeated roots are detected algebraically by the common factors of the polynomial and its formal derivative. The leading coefficient is the coefficient of the highest-degree term; if it equals one the polynomial is called monic. Relations among roots and coefficients are given by symmetric formulas commonly known as Vieta's relations.

Examples and integer clearing

As a concrete demonstration consider an equation written with rational coefficients. For instance, the equality

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

is an algebraic equation over the rational numbers. Multiplying both sides by the least common multiple of the denominators (in this case 42) yields an equivalent polynomial equation with integer coefficients:

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

Passing to integer coefficients in this way is a routine step in many proofs and reductions in elementary number theory and diophantine analysis.

Solution domains and Diophantine problems

When solving an algebraic equation one must specify the domain of allowed solutions. If one seeks integer solutions the problem becomes a Diophantine equation, a class of questions that ranges from elementary to famously difficult. Allowing rational solutions leads to questions about rational points on algebraic curves and higher-dimensional varieties. Over the real numbers one studies real roots and their ordering; over the complex numbers one benefits from algebraic closure, which simplifies existence statements for univariate equations.

Methods of solution: symbolic and numeric

Classical explicit methods exist in low degrees: quadratic equations are solved by completing the square; cubic and quartic equations admit general formulas expressible using radicals, discovered in the Renaissance era by figures such as Gerolamo Cardano and Lodovico Ferrari. For general polynomials of degree five or higher there is no universal formula in radicals. Modern algebra provides structural tools such as Galois theory, which gives criteria for when a polynomial equation can be solved by radicals, a theory developed from the work of Évariste Galois and clarified by later mathematicians including Niels Henrik Abel.

Beyond closed-form formulas, a variety of algorithmic and numerical methods are used: factorization algorithms can decompose polynomials over various fields; resultants and elimination theory reduce multivariate systems to univariate problems; Gröbner bases give systematic ways to handle polynomial ideals and solve systems symbolically; and numerical root-finding methods such as Newton's method approximate real or complex roots to high precision. Additional tests and bounds—such as Descartes' rule of signs, the rational root theorem, and the discriminant—give information about the number, nature and multiplicity of real roots without computing them exactly.

Algebraic geometry viewpoint

Systems of algebraic equations define geometric objects called algebraic varieties: a single polynomial in two variables defines a plane curve, while systems of several polynomials define higher-dimensional loci. This geometric perspective connects algebraic equations to topology and geometry and motivates advanced study in algebraic geometry, where structural questions about solution sets are translated into properties of rings and ideals. Computational algebraic geometry provides tools that combine symbolic and numeric methods to study specific systems arising in engineering, physics and other applied fields.

Historical notes and significance

The history of solving algebraic equations spans many cultures and centuries. Ancient civilizations developed techniques for particular low-degree problems. Renaissance mathematicians found explicit formulas for cubics and quartics, but the discovery that general quintic equations lack a formula in radicals was a turning point: Abel established the impossibility in the early nineteenth century, and Galois theory provided the conceptual explanation linking solvability to group theoretic structure. These developments helped shape modern abstract algebra.

Algebraic equations form a central object of study because they combine explicit computation with deep theoretical structure. They occur throughout pure and applied mathematics, from solving numerical models in engineering to understanding rational points in number theory and the shape of algebraic varieties in geometry.