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In algebraic geometry, an affine variety, or affine algebraic variety, over an algebraically closed field k is the zero-locus in the affine space k of some finite family of polynomials of n variables with coefficients in k that generate a prime ideal. If the condition of generating a prime ideal is removed, such a set is called an algebraic set. A Zariski open subvariety of an affine variety is called a quasi-affine variety.
Some texts do not require a prime ideal, and call irreducible an algebraic variety defined by a prime ideal. This article refers to zero-loci of not necessarily prime ideals as affine algebraic sets.
In some contexts, it is useful to distinguish the field k in which the coefficients are considered, from the algebraically closed field K over which the zero-locus is considered. In this case, the variety is said defined over k, and the points of the variety that belong to k are said k-rational or rational over k. In the common case where k is the field of real numbers, a k-rational point is called a real point. When the field k is not specified, a rational point is a point that is rational over the rational numbers. For example, Fermat's Last Theorem asserts that the affine algebraic variety defined by x + y − 1 = 0 has no rational points for any integer n greater than two.