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In algebraic geometry, an affine algebraic set is the set of the common zeros over an algebraically closed field k of some family of polynomials in the polynomial ring An affine variety or affine algebraic variety, is an affine algebraic set such that the ideal generated by the defining polynomials is prime. Some texts call variety any algebraic set, and irreducible variety an algebraic set whose defining ideal is prime (affine variety in the above sense). In some contexts (see, for example, Hilbert's Nullstellensatz), it is useful to distinguish the field k in which the coefficients are considered, from the algebraically closed field K (containing k) over which the common zeros are considered (that is, the points of the affine algebric set are in Kn). In this case, the variety is said defined over k, and the points of the variety that belong to kn 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 (it is a curve) defined by xn + yn − 1 = 0 has no rational points for any integer n greater than two. An affine algebraic set is the set of solutions in an algebraically closed field k of a system of polynomial equations with coefficients in k. More precisely, if are polynomials with coefficients in k, 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 X is an affine algebraic set, and I is the ideal of all polynomials that are zero on X, 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.

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