A corollary of this theorem is that, if two irreducible polynomials (or more generally two square-free polynomials) define the same hypersurface, then one is the product of the other by a nonzero constant. Hypersurfaces are exactly the subvarieties of dimension of an affine space of dimension of . This is the geometric interpretation of the fact that, in a polynomial ring over a field, the height of an ideal is 1 if and only if the ideal is a principal ideal. In the case of possibly reducible hypersurfaces, this result may be restated as follows: hypersurfaces are exactly the algebraic sets whose all irreducible components have dimension . A "real hypersurface" is a hypersurface that is defined by a polynomial with real coefficients. In this case the algebraically closed field over which the points are defined is generally the field formula_4 of complex numbers. The "real points" of a real hypersurface are the points that belong to formula_5 The set of the real points of a real hypersurface is the "real part" of the hypersurface. Often, it is left to the context whether the term "hypersurface" refers to all points or only to the real part. If the coefficients of the defining polynomial belong to a field that is not algebraically closed (typically the field of rational numbers, a finite field or a number field), one says that the hypersurface is "defined over" , and the points that belong to formula_6 are "rational" over (in the case of the field of rational numbers, "over " is generally omitted). For example, the imaginary -sphere defined by the equation is a real hypersurface without any real point, which is defined over the rational numbers. It has no rational point, but has many points that are rational over the Gaussian rationals. A of dimension in a projective space of dimension over a field is defined by a homogeneous polynomial formula_8 in indeterminates. As usual, means that all monomials of have the same degree, or, equivalently that formula_9 for every constant , where is the degree of the polynomial. The of the hypersurface are the points of the projective space whose projective coordinates are zeros of . If one chooses the hyperplane of equation formula_10 as hyperplane at infinity, the complement of this hyperplane is an affine space, and the points of the projective hypersuface that belong to this affine space form an affine hypersurface of equation formula_11 Conversely, given an affine hypersurface of equation formula_2 it defines a projective hypersurface, called its , whose equation is obtained by homogenizing .