Editing Hilbert's Nullstellensatz (section)

== Formulation ==
Let <math>k</math> be a [[field (mathematics)|field]] (such as the [[rational number]]s) and <math>K</math> be an algebraically closed [[field extension]] of <math>k</math> (such as the [[complex number]]s). Consider the [[polynomial ring]] <math>k[X_1, \ldots, X_n]</math> and let <math>I</math> be an [[Ideal (ring theory)|ideal]] in this ring. The [[algebraic set]] <math>\mathrm V(I)</math> defined by this ideal consists of all <math>n</math>-tuples <math>\mathbf x = (x_1, \dots, x_n)</math> in <math>K^n</math> such that <math>f(\mathbf x) = 0</math> for all <math>f</math> in {{awrap|<math>I</math>.}} Hilbert's Nullstellensatz states that if ''p'' is some polynomial in <math>k[X_1, \ldots, X_n]</math> that vanishes on the algebraic set <math>\mathrm V(I)</math>, i.e. <math>p(\mathbf x) = 0</math> for all <math>\mathbf x</math> in <math>\mathrm V(I)</math>, then there exists a [[natural number]] <math>r</math> such that <math>p^r</math> is in <math>I</math>.<ref>{{harvnb|Zariski–Samuel|loc=Ch. VII, Theorem 14|last1=Zariski|year=1960}}.</ref>

An immediate corollary is the '''weak Nullstellensatz''': The ideal <math>I \subseteq k[X_1, \ldots, X_n]</math> contains 1 if and only if the polynomials in <math>I</math> do not have any common zeros in ''K<sup>n</sup>''. Specializing to the case <math>k=K=\mathbb{C}, n=1</math>, one immediately recovers a restatement of the [[fundamental theorem of algebra]]: a polynomial ''P'' in <math>\mathbb{C}[X]</math> has a root in <math>\mathbb{C}</math> if and only if deg ''P'' ≠ 0.  For this reason, the (weak) Nullstellensatz has been referred to as a generalization of the fundamental theorem of algebra for multivariable polynomials.<ref>{{Cite book |last=Cox |first=David A. |url=https://link.springer.com/10.1007/978-3-319-16721-3 |title=Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra |last2=Little |first2=John |last3=O’Shea |first3=Donal |date=2015 |publisher=Springer International Publishing |isbn=978-3-319-16720-6 |series=Undergraduate Texts in Mathematics |location=Cham |language=en |doi=10.1007/978-3-319-16721-3}}</ref> The weak Nullstellensatz may also be formulated as follows: if ''I'' is a proper ideal in <math>k[X_1, \ldots, X_n],</math> then V(''I'') cannot be [[empty set|empty]], i.e. there exists a common zero for all the polynomials in the ideal in every algebraically closed extension of ''k''.  This is the reason for the name of the theorem, the full version of which can be proved easily from the 'weak' form using the [[Rabinowitsch trick]]. The assumption of considering common zeros in an algebraically closed field is essential here; for example, the elements of the proper ideal (''X''<sup>2</sup> + 1) in <math>\R[X]</math> do not have a common zero in <math>\R.</math>

With the notation common in algebraic geometry, the Nullstellensatz can also be formulated as

:<math>\hbox{I}(\hbox{V}(J))=\sqrt{J}</math>

for every ideal ''J''. Here, <math>\sqrt{J}</math> denotes the [[radical of an ideal|radical]] of ''J'' and I(''U'') is the ideal of all polynomials that vanish on the set ''U''.

In this way, taking <math>k = K</math> we obtain an order-reversing [[bijective]] correspondence between the algebraic sets in ''K''<sup>''n''</sup> and the [[radical ideal]]s of <math>K[X_1, \ldots, X_n].</math> In fact, more generally, one has a [[Galois connection]] between subsets of the space and subsets of the algebra, where "[[Zariski closure]]" and "radical of the ideal generated" are the [[closure operator]]s.

As a particular example, consider a point <math>P = (a_1, \dots, a_n) \in K^n</math>. Then <math>I(P) = (X_1 - a_1, \ldots, X_n - a_n)</math>. More generally,

:<math>\sqrt{I} = \bigcap_{(a_1, \dots, a_n) \in V(I)}  (X_1 - a_1, \dots, X_n - a_n).</math>

Conversely, every [[maximal ideal]] of the polynomial ring <math>K[X_1,\ldots,X_n]</math> (note that <math>K</math> is algebraically closed) is of the form <math>(X_1 - a_1, \ldots, X_n - a_n)</math> for some <math>a_1,\ldots,a_n \in K</math>.

As another example, an algebraic subset ''W'' in ''K''<sup>''n''</sup> is [[irreducible space|irreducible]] (in the Zariski topology) if and only if <math>I(W)</math> is a prime ideal.