algébrique

Algébrique, in mathematics, refers to concepts that can be described by polynomial equations. An object is said to be algebraic over a field F if it is a root of a nonzero polynomial with coefficients in F. If every element of an extension field E is algebraic over F, the extension is called algebraic; otherwise it is transcendental. The term underlies several central notions.

Algebraic numbers are elements that are algebraic over the rational numbers, equivalently roots of polynomials with integer coefficients. The field of algebraic numbers is the union of all finite extensions of Q inside the complex numbers; it is countable and has rich algebraic structure.

Algebraic geometry studies geometric objects defined by polynomial equations. An algebraic variety over a field is the set of common zeros of one or more polynomials. These varieties can be studied in affine or projective space, and they carry geometric properties that reflect algebraic relations among the defining equations. The foundational language is that of commutative algebra, with concepts such as ideals, coordinate rings, and morphisms.

Algebraic functions are functions that satisfy a polynomial equation whose coefficients lie in a given field. For example, any root of x^2-2 or y^2 = x defines an algebraic relation. The contrast with transcendental objects—such as most analytic functions like exp or pi—is a common distinction in number theory and analysis.

In French mathematical usage, algébrique also appears in areas such as géométrie algébrique to denote the algebraic origin of the objects and methods involved.