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Theorems
Hilbert's Nullstellensatz underlies at the most fundamental level.

The theorem was originally proved by Hilbert for the purposes of.

Naively, one can think of the nullstellensatz as a "completeness theorem for polynomial arithmetic".

We give several different presentaions:


 * The usual statement, and several immediate corollaries that illustrate the power of the theorem
 * A stronger, commutative algebraic statement due to Bourbaki
 * A simple statement of a so-called formal nullstellensatz

As well as two different proofs: those of Bourbaki and Artin-Tate.

First we describe some conventions for notation used in this article:


 * $k$ is an algebraically closed field


 * $\mathcal A = k[x_1,\ldots,x_n]$ is the polynomial ring in $n$ indeterminates


 * $X \subseteq \mathbb A^n_k$ is an affine algebraic set over $k$


 * To each $X$ we associate an ideal


 * $\displaystyle I(X) = \{ f \in A : f(x)=0\ \forall x \in X \}\subseteq \mathcal A$


 * To an ideal $I \subseteq \mathcal A$ we associate the zero locus


 * $Z(I) = \{ x \in \mathbb A^n_k : f(x)=0\ \forall x \in I \}$


 * $\operatorname{rad}(I)$ denotes the radical of an ideal $I$


 * $A(X) = \mathcal A / I(X)$ is the affine coordinate ring of $X$


 * For polynomials $g_1,\ldots, g_d \in \mathcal A$, $\langle g_1, \ldots, g_d \rangle \subseteq \mathcal A$ denotes the ideal generated by the $g_i$

Usual Statement (1)

 * $\displaystyle I(Z(I)) = \operatorname{rad}(I)$

Corollary (1.1)
There is a one-to-one, inclusion reversing correspondence between affine algebraic sets in $\mathbb A^n_k$ and radical ideals in $A$.

Corollary (1.2)
Let


 * $ f_1 ( x_1, \ldots, x_n ) = 0$
 * $\qquad \vdots$
 * $ f_r ( x_1, \ldots, x_n ) = 0$

be a system of polynomial equations over $k$.

This system has no solution in $k^n$ if and only if there exist polynomials $p_i \in \mathcal A$ such that


 * $\displaystyle \sum_{i = 1}^r p_if_i = 1$

Remark
For this criterion to be useful, one requires bounds on the degrees of the $p_i$, but such bounds exceed the scope of this article.

Corollary (1.3)
Let $\mathcal B$ be a $k$-algebra.

Then $\mathcal B$ is the affine coordinate ring of some $X$ if and only if $\mathcal B$ is reduced and finitely generated (as a $k$-algebra).

Corollary (1.4)
Let $\mathfrak m$ be a maximal ideal of $A(X)$.

Then there exists a point $P = (a_1, \ldots, a_n) \in \mathbb A^n$ of affine $n$-space such that


 * $\displaystyle \mathfrak m = \left\langle x_1-a_1,\ldots, x_n-a_n \right\rangle$

That is, there is a one-to-one correspondence between ideals of $A(X)$ and points of $X$.

Corollary (1.5)
The category of reduced affine algebraic sets over $k$ and morphisms of affine algebraic sets is equivalent to the category of reduced, finitely generated $k$-algebras with the arrows reversed.

Projective Nullstellensatz (2)
Projective algebraic sets can be thought of as Conical sets in affine $(n+1)$-space.

Conversely, provided $k$ is infinite, the ideal associated to such an affine set is homogeneous.

In this way the projective nullstellensatz is a special case of the usual (affine) nullstellensatz.

Statement
Let $J \subseteq k[x_0,\ldots, x_n]$ be a homogeneous ideal such that $J \subseteq \langle x_0,\ldots,x_n \rangle \subseteq $.

We have the zero locus of $J$


 * $\displaystyle Z(J) = \left\{ (a_0:\cdots : a_n) \in \mathbb P^n_k : f(a_0,\ldots, a_n)=0\ \forall f \in J \right\}$

Let $Y \subseteq \mathbb P^n$ be a projective algebraic set.

We define $J(Y)$ to be the ideal generated by homogeneous polynomials of positive degree such that $f(y) = 0$ for all $y \in Y$.

Then


 * $ J(V(J)) = \operatorname{rad}(J)$

General Nullstellensatz (3)
Let $R$ be a Jacobson ring.

Let $S$ be a finitely generated $R$-algebra.

Let $\mathfrak m \subseteq S$ be a maximal ideal.

Then:
 * $S$ is a Jacobson ring
 * $\mathfrak m \cap R$ is a maximal ideal of $R$
 * $S/\mathfrak m$ is a finite field extension of $R/(\mathfrak m \subseteq S)$

Formal Nullstellensatz
Let $\left( \mathscr X, \leq \right)$ and $\left( \mathscr Y, \leq \right)$ be Partially ordered sets.

Let $J: \mathscr X \to \mathscr Y$, $Z: \mathscr Y \to \mathscr X$ be functions such that
 * $x, y \in \mathscr X$ and $x \leq y$ implies that $J(x) \geq J(x)$
 * $i, j \in \mathscr Y$ and $i \leq j$ implies that $Z(i) \geq Z(j)$
 * $x \in \mathscr X$ implies that $\left( Z \circ J \right)(x) \geq x$
 * $i \in \mathscr Y$ implies that $\left( J \circ Z \right)(i) \geq i$

Then $J$ and $Z$ establish a one-to-one correspondence between the subsets


 * $\left\{ J(\mathscr A) : \mathscr A \subseteq \mathscr X \right\}, \qquad \left\{ Z(\mathscr B): \mathscr B \subseteq \mathscr Y \right\}$