algebraische

Algebraische is the German adjective meaning "algebraic." In mathematics, it is used to describe objects defined by algebraic relations, typically polynomial equations with coefficients in a field. The term appears across several subfields and denotes properties or objects that can be expressed without invoking transcendental functions.

Algebraic numbers are complex numbers that are roots of nonzero polynomials with rational coefficients. For example, sqrt(2) is algebraic, as it satisfies x^2 - 2 = 0; the golden ratio (1+sqrt(5))/2 is algebraic, satisfying x^2 - x - 1 = 0. By contrast, numbers such as pi or e are transcendental, not algebraic.

In algebraic geometry, algebraische varieties are the solution sets to systems of polynomial equations in affine or projective space. These objects are defined purely by polynomials, in contrast to analytic or differential-geometric objects.

In field theory and ring theory, an element is called algebraisch over a field if it is a root of a nonzero polynomial with coefficients in that field. The minimal polynomial and the degree of the extension measure the algebraic complexity.

The term also appears in broader German mathematical language, for example algebraische Gleichungen (algebraic equations) and algebraische Funktionen (algebraic functions). In English-language contexts, the corresponding term is algebraic.