Affine variety

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 ${\displaystyle k[X_{1},\ldots ,X_{n}].}$ 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 use the term variety for 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 algebraic 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.^[1] 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.