# Algebraic Variety

Algebraic Variety

In mathematics, an algebraic set is the set of solutions of a system of polynomial equations. Algebraic sets are sometimes also called algebraic varieties, but normally an algebraic variety is an irreducible algebraic set, i.e. one which is not the union of two other algebraic sets. Algebraic sets and algebraic varieties are the central objects of study in algebraic geometry.

The notion of variety is similar to that of manifold, the difference being that a variety may have singular points, while a manifold may not. In many languages, both varieties and manifolds are named by the same word.

Proven around the year 1800, the fundamental theorem of algebra establishes a link between algebra and geometry by showing that a monic polynomial in one variable with complex coefficients (an algebraic object) is determined by the set of its roots (a geometric object). Generalizing this result, Hilbert's Nullstellensatz provides a fundamental correspondence between ideals of polynomial rings and algebraic sets. Using the Nullstellensatz and related results, mathematicians have established a strong correspondence between questions on algebraic sets and questions of ring theory. This correspondence is the specifity of algebraic geometry among the other subareas of geometry.

### Other articles related to "algebraic, variety, algebraic variety":

Abstract Variety
... In mathematics, in the field of algebraic geometry, the idea of abstract variety is to define a concept of algebraic variety in an intrinsic way ... Weil, in his foundational work, who gave a first acceptable definition of algebraic variety that stood outside projective space ... The simplest notion of algebraic variety is affine algebraic variety ...
Algebraic Variety - Algebraic Manifolds
... An algebraic manifold is an algebraic variety which is also an m-dimensional manifold, and hence every sufficiently small local patch is isomorphic to km ... Equivalently, the variety is smooth (free from singular points) ... When k is the real numbers, R, algebraic manifolds are called Nash manifolds ...
Abstract Algebraic Variety - Introduction and Definitions - Abstract Varieties
... In classical algebraic geometry, all varieties were by definition quasiprojective varieties, meaning that they were open subvarieties of closed subvarieties of projective space ... For example, in Chapter 1 of Hartshorne a variety over an algebraically closed field is defined to be a quasi-projective variety, but from Chapter 2 onwards, the term variety (also called an abstract variety ... So classically the definition of an algebraic variety required an embedding into projective space, and this embedding was used to define the topology on the variety and the regular functions on the ...
Ring (mathematics) - Advanced Examples - Function Field of An Irreducible Algebraic Variety
... To any irreducible algebraic variety is associated its function field ... The points of an algebraic variety correspond to valuation rings contained in the function field and containing the coordinate ring ... The study of algebraic geometry makes heavy use of commutative algebra to study geometric concepts in terms of ring-theoretic properties ...
Abstract Algebraic Variety
... In mathematics, algebraic varieties (also called varieties) are one of the central objects of study in algebraic geometry ... Classically, an algebraic variety was defined to be the set of solutions of a system of polynomial equations, over the real or complex numbers ... Modern definitions of an algebraic variety generalize this notion in several different ways, while attempting to preserve the geometric intuition behind the original definition ...

### Famous quotes containing the words variety and/or algebraic:

Gradually we come to admit that Shakespeare understands a greater extent and variety of human life than Dante; but that Dante understands deeper degrees of degradation and higher degrees of exaltation.
—T.S. (Thomas Stearns)

I have no scheme about it,—no designs on men at all; and, if I had, my mode would be to tempt them with the fruit, and not with the manure. To what end do I lead a simple life at all, pray? That I may teach others to simplify their lives?—and so all our lives be simplified merely, like an algebraic formula? Or not, rather, that I may make use of the ground I have cleared, to live more worthily and profitably?
Henry David Thoreau (1817–1862)