algebraischer

Algebraischer is the German adjective that denotes a relation to or a definition by algebra. It is used across various branches of mathematics to indicate that an object, property, or construction is given by algebraic relations, typically polynomial equations with coefficients in a specified field. The term helps distinguish algebraic phenomena from those that are analytic, geometric in a non-algebraic sense, or transcendental.

Key concepts often described as algebraisch include algebraische Zahlen, algebraische Mengen, algebraische Erweiterungen and algebraische Gruppen. An algebraische Zahl is a complex or real number that is a root of a nonzero polynomial with integer (or, more generally, rational) coefficients. An algebraische Menge (or algebraische Varietät) is a set of common zeros of a family of polynomials over a field, central in algebraic geometry. An algebraische Erweiterung of a field is a field extension in which every element is a root of a polynomial with coefficients in the base field. An algebraische Gruppe refers to a group defined by polynomial equations in the language of algebraic geometry.

The term also appears in everyday mathematical language: an algebraischer Ausdruck (an algebraic expression) is built from variables, constants and algebraic operations such as addition, multiplication and exponentiation by natural numbers, as opposed to purely numerical or transcendental expressions. The distinction between algebraisch and transcendental is a recurring theme in number theory and field theory, illustrating whether elements satisfy polynomial relations.