In algebraic geometry, an affine variety, or affine algebraic variety, over an algebraically closed field is the zero-locus in the affine space of some finite family of polynomials of variables with coefficients in that generate a prime ideal. If the condition of generating a prime ideal is removed, such a set is called an (affine) 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 in which the coefficients are considered, from the algebraically closed field (containing ) over which the zero-locus is considered (that is, the points of the affine variety are in ). In this case, the variety is said defined over , and the points of the variety that belong to are said -rational or rational over . In the common case where is the field of real numbers, a -rational point is called a real point. When the field 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 (it is a curve) defined by has no rational points for any integer greater than two. Introduction An affine algebraic set is the set of solutions in an algebraically closed field of a system of polynomial equations with coefficients in . More precisely, if are polynomials with coefficients in , they define an affine algebraic set An affine (algebraic) variety is an affine algebraic set which is not the union of two proper affine algebraic subsets. Such an affine algebraic set is often said to be irreducible. If is an affine algebraic set, and is the ideal of all polynomials that are zero on , then the quotient ring is called the of X. If X is an affine variety, then I is prime, so the coordinate ring is an integral domain. The elements of the coordinate ring R are also called the regular functions or the polynomial functions on the variety. They form the ring of regular functions on the variety, or, simply, the ring of the variety; in other words (see #Structure sheaf), it is the space of global sections