Editing Polynomial ring (section)

===Bézout's theorem===
{{main|Bézout's theorem}}
Bézout's theorem may be viewed as a multivariate generalization of the version of the [[fundamental theorem of algebra]] that asserts that a univariate polynomial of degree {{mvar|n}} has {{mvar|n}} complex roots, if they are counted with their multiplicities.

In the case of [[bivariate polynomial]]s, it states that two polynomials of degrees {{mvar|d}} and {{mvar|e}} in two variables, which have no common factors of positive degree, have exactly {{mvar|de}} common zeros in an [[algebraically closed field]] containing the coefficients, if the zeros are counted with their multiplicity and include the [[point at infinity|zeros at infinity]].

For stating the general case, and not considering "zero at infinity" as special zeros, it is convenient to work with [[homogeneous polynomial]]s, and consider zeros in a [[projective space]]. In this context, a ''projective zero'' of a homogeneous polynomial <math>P(X_0, \ldots, X_n)</math> is, up to a scaling, a {{math|(''n'' + 1)}}-[[tuple]] <math>(x_0, \ldots, x_n)</math>  of elements of {{mvar|K}} that is different from {{math|(0, …, 0)}}, and such that <math>P(x_0, \ldots, x_n) = 0 </math>. Here, "up to a scaling" means that <math>(x_0, \ldots, x_n)</math> and <math>(\lambda x_0, \ldots, \lambda x_n)</math> are considered as the same zero for any nonzero <math>\lambda\in K.</math> In other words, a zero is a set of [[homogeneous coordinates]] of a point in a projective space of dimension {{mvar|n}}.

Then, Bézout's theorem states: Given {{mvar|n}} homogeneous polynomials of degrees <math>d_1, \ldots, d_n</math> in {{math|''n'' + 1}} indeterminates, which have only a finite number of common projective zeros in an [[algebraically closed extension]] of {{mvar|K}}, the sum of the [[multiplicity (mathematics)#Intersection multipliicty|multiplicities]] of these zeros is the product <math>d_1 \cdots d_n.</math>