Editing Hilbert's Nullstellensatz (section)

== Proofs ==
There are many known proofs of the theorem. Some are [[constructive proof|non-constructive]], such as the first one. Others are constructive, as based on [[algorithm]]s for expressing {{math|1}} or {{math|''p{{sup|r}}''}} as a [[linear combination]] of the generators of the ideal.

===Using Zariski's lemma===
[[Zariski's lemma]] asserts that if a field is [[finitely generated algebra|finitely generated]] as an [[associative algebra]] over a field ''K'', then it is a [[finite field extension]] of ''K'' (that is, it is also finitely generated as a [[vector space]]).  If <math>\mathfrak{m}</math> is a maximal ideal of <math>K[X_1,\ldots,X_n]</math> for algebraically closed ''K'', then Zariski's lemma implies that <math>K[X_1,\ldots,X_n]/ \mathfrak{m}</math> is a finite field extension of ''K'', and thus, by algebraic closure, must be ''K''.  From this, it follows that there is an <math>a = (a_1,\dots,a_n)\in K^n</math> such that <math>X_i-a_i\in\mathfrak{m}</math> for <math>i=1,\dots, n</math>.  In other words, 
:<math>\mathfrak{m} \supseteq \mathfrak{m}_a=(X_1 - a_1, \ldots, X_n - a_n)</math>
for some <math>a = (a_1,\dots,a_n)\in K^n</math>.  But <math>\mathfrak{m}_a</math> is clearly maximal, so <math>\mathfrak{m}=\mathfrak{m}_a</math>.  This is the weak Nullstellensatz: every maximal ideal of <math>K[X_1,\ldots,X_n]</math> for algebraically closed ''K'' is of the form <math>\mathfrak{m}_a=(X_1 - a_1, \ldots, X_n - a_n)</math> for some <math>a = (a_1,\dots,a_n)\in K^n</math>.  Because of this close relationship, some texts refer to Zariski's lemma as the weak Nullstellensatz or as the 'algebraic version' of the weak Nullstellensatz.<ref>{{Cite book |last=Patil |first=Dilip P. |title=Introduction to Algebraic Geometry and Commutative Algebra |last2=Storch |first2=Uwe |publisher=World Scientific |year=2010 |isbn=978-9814307581}}</ref><ref>{{Cite book |last=Reid |first=Miles |title=Undergraduate commutative algebra |date=1995 |publisher=Cambridge University Press |isbn=978-0-521-45255-7 |series=London Mathematical Society student texts |location=Cambridge ; New York}}</ref> 

The full Nullstellensatz can also be proved directly from Zariski's lemma without employing the Rabinowitsch trick. Here is a sketch of a proof using this lemma.<ref>{{harvnb|Atiyah–Macdonald|loc=Ch. 7}}, Exercise 7.14.</ref>

Let <math>A = K[X_1, \ldots, X_n]</math> (''K'' an algebraically closed field), ''J'' an ideal of ''A,'' and <math>V=\mathrm{V}(J)</math> the common zeros of ''J'' in <math>K^n</math>. Clearly, <math>\sqrt{J} \subseteq \mathrm{I}(V)</math>, where <math>\mathrm{I}(V)</math> is the ideal of polynomials in ''A'' vanishing on ''V''. To show the opposite inclusion, let <math>f \not\in \sqrt{J}</math>. Then <math>f \not\in \mathfrak{p}</math> for some prime ideal <math>\mathfrak{p} \supseteq J</math> in ''A''. Let <math>R = (A/\mathfrak{p}) [1/\bar{f}]</math>, where <math>\bar{f}</math>is the image of ''f'' under the natural map <math>A \to A/\mathfrak{p}</math>, and <math>\mathfrak{m}</math> be a maximal ideal in ''R''. By Zariski's lemma, <math>R/\mathfrak{m}</math> is a finite extension of ''K'', and thus, is ''K'' since ''K'' is algebraically closed. Let <math>x_i</math> be the images of <math>X_i</math> under the natural map <math>A \to A/\mathfrak{p}\to R \to R/\mathfrak{m}\cong K</math>. It follows that, by construction, <math>x = (x_1, \ldots, x_n) \in V</math> but <math>f(x) \ne 0</math>, so <math>f \notin \mathrm{I}(V)</math>.

===Using resultants===
The following constructive proof of the weak form is one of the oldest proofs (the strong form results from the [[Rabinowitsch trick]], which is also constructive).

The [[resultant]] of two polynomials depending on a variable {{mvar|x}} and other variables is a polynomial in the other variables that is in the ideal generated by the two polynomials, and has the following properties: if one of the polynomials is [[monic polynomial|monic]] in {{mvar|x}}, every zero (in the other variables) of the resultant may be extended into a common zero of the two polynomials.

The proof is as follows.

If the ideal is [[principal ideal|principal]], generated by a non-constant polynomial {{mvar|p}} that depends on {{mvar|x}}, one chooses arbitrary values for the other variables. The [[fundamental theorem of algebra]] asserts that this choice can be extended to a zero of {{mvar|p}}.

In the case of several polynomials <math>p_1,\ldots, p_n,</math> a linear change of variables  allows to suppose that <math>p_1</math> is monic in the first variable {{mvar|x}}. Then, one introduces <math>n-1</math> new variables <math>u_2, \ldots, u_n,</math> and one considers the resultant
:<math>R=\operatorname{Res}_x(p_1,u_2p_2+\cdots +u_np_n).</math>

As {{mvar|R}} is in the ideal generated by <math>p_1,\ldots, p_n,</math> the same is true for  the coefficients in {{mvar|R}} of the [[monomial]]s in <math>u_2, \ldots, u_n.</math> So, if {{math|1}} is in the ideal generated by these coefficients, it is also in the ideal generated by <math>p_1,\ldots, p_n.</math> On the other hand, if these coefficients have a common zero, this zero can be extended to a common zero of <math>p_1,\ldots, p_n,</math> by the above property of the resultant.

This proves the weak Nullstellensatz by induction on the number of variables.

=== Using Gröbner bases ===

A [[Gröbner basis]] is an algorithmic concept that was introduced in 1973 by [[Bruno Buchberger]]. It is presently fundamental in [[computational geometry]]. A Gröbner basis is a special generating set of an ideal from which most properties of the ideal can easily be extracted. Those that are related to the Nullstellensatz are the following:
*An ideal contains {{math|1}} if and only if its [[reduced Gröbner basis]] (for any [[monomial ordering]]) is {{math|1}}.
*The number of the common zeros of the polynomials in a Gröbner basis is strongly related to the number of [[monomial]]s that are [[Gröbner basis#Reduction|irreducible]]s by the basis. Namely, the number of common zeros is infinite if and only if the same is true for the irreducible monomials; if the two numbers are finite, the number of irreducible monomials equals the numbers of zeros (in an algebraically closed field), counted with multiplicities.
*With a [[lexicographic order|lexicographic monomial order]], the common zeros can be computed by solving iteratively [[univariate polynomial]]s (this is not used in practice since one knows better algorithms).
* Strong Nullstellensatz: a power of {{mvar|p}} belongs to an ideal {{mvar|I}} if and only the [[Gröbner basis#Saturation|saturation]] of {{mvar|I}} by {{mvar|p}} produces the Gröbner basis {{math|1}}. Thus, the strong Nullstellensatz results almost immediately from the definition of the saturation.