Editing Polynomial ring (section)

=== Hilbert's Nullstellensatz ===
{{Main|Hilbert's Nullstellensatz}}
The Nullstellensatz (German for "zero-locus theorem") is a theorem, first proved by [[David Hilbert]], which extends to the multivariate case some aspects of the [[fundamental theorem of algebra]]. It is foundational for [[algebraic geometry]], as establishing a strong link between the algebraic properties of <math>K[X_1, \ldots, X_n]</math> and the geometric properties of [[algebraic varieties]], that are (roughly speaking) set of points defined by [[implicit equation|implicit polynomial equations]].

The Nullstellensatz, has three main versions, each being a corollary of any other. Two of these versions are given below. For the third version, the reader is referred to the main article on the Nullstellensatz.

The first version generalizes the fact that a nonzero univariate polynomial has a [[complex number|complex]] zero if and only if it is not a constant. The statement is: ''a set of polynomials {{mvar|S}} in <math>K[X_1, \ldots, X_n]</math> has a common zero in an [[algebraically closed field]] containing {{mvar|K}}, if and only if'' {{math|1}} ''does not belong to the [[ideal (ring theory)|ideal]] generated by {{mvar|S}}, that is, if'' {{math|1}} ''is not a [[linear combination]] of elements of {{mvar|S}} with polynomial coefficients''.

The second version generalizes the fact that the [[irreducible polynomial|irreducible univariate polynomial]]s over the complex numbers are [[associate elements|associate]] to a polynomial of the form <math>X-\alpha.</math> The statement is: ''If {{mvar|K}} is algebraically closed, then the [[maximal ideal]]s of <math>K[X_1, \ldots, X_n]</math> have the form <math>\langle X_1 - \alpha_1, \ldots, X_n - \alpha_n \rangle.</math>''