Algebraic variety

:This article is about algebraic varieties. For varieties of algebras, and an explanation of the difference, see variety (universal algebra).

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An affine algebraic variety is essentially the set of common zeroes of a set of polynomials, and is one of the central objects of study in classical (and to some extent, modern) algebraic geometry. Historically, the fundamental theorem of algebra established a link between algebra and geometry by saying that a polynomial in one variable over the complex numbers is determined by the set of its roots, which is an inherently geometric object. Building on this result, Hilbert's Nullstellensatz provides a fundamental correspondence between ideals of polynomial rings and subsets of affine space. Using the Nullstellensatz and related results, we are able to capture the geometric notion of a variety in algebraic terms as well as bring geometry to bear on questions of ring theory.

Related Topics:
Algebraic geometry - Fundamental theorem of algebra - Polynomial - Complex numbers - Nullstellensatz - Polynomial ring - Ring theory

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