Algebraic

Algebraic is an adjective used in mathematics to describe objects that can be defined by, or are described by, polynomial equations with coefficients in a given field. An object is called algebraic over a field F if it is a root of some nonzero polynomial with coefficients in F. The term contrasts with transcendental, which are not roots of any such polynomial.

Algebraic numbers are the roots of nonzero polynomials with integer coefficients (equivalently, rational coefficients). They form a countable field extension of the rationals; real and complex numbers such as sqrt(2), the cube root of 2, and the golden ratio are algebraic, while most real numbers are not. An algebraic function is a function that satisfies a polynomial equation in its variable with coefficients from a given field; for example, y = sqrt(x) satisfies y^2 - x = 0 and is algebraic over the field of real rational functions.

In algebraic geometry, "algebraic" describes objects defined by polynomial equations, notably algebraic varieties defined as common zeros of polynomials in several variables. The study centers on rings of polynomials, coordinate rings, and morphisms between varieties. More broadly, algebraic objects include groups, rings, and fields described by polynomial identities, as in algebraic groups or algebraic extensions.

Algebraic notions play roles in number theory, geometry, and logic, and are often contrasted with analytic or transcendental concepts. The term also appears in algebraic closure and algebraic independence, among other contexts.