of the structure sheaf of X. The dimension of a variety is an integer associated to every variety, and even to every algebraic set, whose importance relies on the large number of its equivalent definitions (see Dimension of an algebraic variety). Examples The complement of a hypersurface in an affine variety (that is for some polynomial ) is affine. Its defining equations are obtained by saturating by the defining ideal of . The coordinate ring is thus the localization . In particular, (the affine line with the origin removed) is affine. On the other hand, (the affine plane with the origin removed) is not an affine variety; cf. Hartogs' extension theorem. The subvarieties of codimension one in the affine space are exactly the hypersurfaces, that is the varieties defined by a single polynomial. The normalization of an irreducible affine variety is affine; the coordinate ring of the normalization is the integral closure of the coordinate ring of the variety. (Similarly, the normalization of a projective variety is a projective variety.) Rational points For an affine variety over an algebraically closed field , and a subfield of , a -rational point of is a point That is, a point of whose coordinates are elements of . The collection of -rational points of an affine variety is often denoted Often, if the base field is the complex numbers , points which are -rational (where is the real numbers) are called real points of the variety, and -rational points ( the rational numbers) are often simply called rational points. For instance, is a -rational and an -rational point of the variety as it is in and all its coordinates are integers. The point is a real point of that is not -rational, and is a point of that is not -rational. This variety is called a circle, because the set of its -rational points is the unit circle. It has infinitely many -rational points that are the points where is a rational number. The circle is an example of an algebraic curve of degree two that has no -rational point. This can be deduced from the fact that, modulo , the sum of two squares cannot be . It can be proved that an algebraic curve of degree two with a -rational point has infinitely many other -rational points; each such point is the second intersection point of the curve and a line