There are many known proofs of the theorem. Some are
non-constructive, such as the first one. Others are constructive, as based on
algorithms for expressing or 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 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
K is an algebraically closed field and \mathfrak{m} is a maximal ideal of the ring of polynomials K[X_1,\ldots,X_n], then Zariski's lemma implies that K[X_1,\ldots,X_n]/ \mathfrak{m} is a finite field extension of
K, and thus, by algebraic closure, must be
K. From this, it follows that there is an a = (a_1,\dots,a_n)\in K^n such that X_i-a_i\in\mathfrak{m} for i=1,\dots, n. In other words, :\mathfrak{m} \supseteq \mathfrak{m}_a=(X_1 - a_1, \ldots, X_n - a_n) for some a = (a_1,\dots,a_n)\in K^n. But \mathfrak{m}_a is clearly maximal, so \mathfrak{m}=\mathfrak{m}_a. This is the weak Nullstellensatz: every maximal ideal of K[X_1,\ldots,X_n] for algebraically closed
K is of the form \mathfrak{m}_a=(X_1 - a_1, \ldots, X_n - a_n) for some a = (a_1,\dots,a_n)\in K^n. 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. 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. Let A = K[X_1, \ldots, X_n] for algebraically closed field
K, and let
J be an ideal of
A and V=\mathrm{V}(J) be the common zeros of
J in K^n. Recall that \mathrm{I}(V) is radical, and therefore \sqrt{J} \subseteq \mathrm{I}(V), where \mathrm{I}(V) is the ideal of polynomials in
A vanishing on
V. To show the opposite inclusion, let f \not\in \sqrt{J}. Then f \not\in \mathfrak{p} for some prime ideal \mathfrak{p} \supseteq J in
A. Let R = (A/\mathfrak{p}) [1/\bar{f}], where \bar{f}is the image of
f under the natural map A \to A/\mathfrak{p}, and \mathfrak{m} be a maximal ideal in
R. By Zariski's lemma, R/\mathfrak{m} is a finite extension of
K, and thus, is
K since
K is algebraically closed. Let x_i be the images of X_i under the natural map A \to A/\mathfrak{p}\to R \to R/\mathfrak{m}\cong K. It follows that, by construction, x = (x_1, \ldots, x_n) \in V but f(x) \ne 0, so f \notin \mathrm{I}(V).
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 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 in , 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, generated by a non-constant polynomial that depends on , one chooses arbitrary values for the other variables. The
fundamental theorem of algebra asserts that this choice can be extended to a zero of . In the case of several polynomials p_1,\ldots, p_n, a linear change of variables allows to suppose that p_1 is monic in the first variable . Then, one introduces n-1 new variables u_2, \ldots, u_n, and one considers the resultant :R=\operatorname{Res}_x(p_1,u_2p_2+\cdots +u_np_n). As is in the ideal generated by p_1,\ldots, p_n, the same is true for the coefficients in of the
monomials in u_2, \ldots, u_n. So, if is in the ideal generated by these coefficients, it is also in the ideal generated by p_1,\ldots, p_n. On the other hand, if these coefficients have a common zero, this zero can be extended to a common zero of p_1,\ldots, p_n, 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 if and only if its
reduced Gröbner basis (for any
monomial ordering) is . • The number of the common zeros of the polynomials in a Gröbner basis is strongly related to the number of
monomials that are
irreducibles 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 monomial order, the common zeros can be computed by solving iteratively
univariate polynomials (this is not used in practice since one knows better algorithms). • Strong Nullstellensatz: a power of belongs to an ideal if and only the
saturation of by produces the Gröbner basis . Thus, the strong Nullstellensatz results almost immediately from the definition of the saturation. == Generalizations ==