# Hilbert’s Nullstellensatz

Let $K$ be an algebraically closed field, and let $I$ be an ideal in $K[x_{1},\ldots,x_{n}]$, the polynomial ring in $n$ indeterminates.

Define $V(I)$, the of $I$, by

 $V(I)=\{(a_{1},\ldots,a_{n})\in K^{n}\mid f(a_{1},\ldots,a_{n})=0\text{ for all% }f\in I\}$

Weak Nullstellensatz:
If $V(I)=\emptyset$, then $I=K[x_{1},\ldots,x_{n}]$. In other words, the zero set of any proper ideal of $K[x_{1},\ldots,x_{n}]$ is nonempty.

Hilbert’s (Strong) Nullstellensatz:
Suppose $f\in K[x_{1},\ldots,x_{n}]$ satisfies $f(a_{1},\ldots,a_{n})=0$ for every $(a_{1},\ldots,a_{n})\in V(I)$. Then $f^{r}\in I$ for some integer $r>0$.

In the of algebraic geometry, the latter result is equivalent to the statement that $\operatorname{Rad}(I)=I(V(I))$, that is, the radical of $I$ is equal to the ideal of $V(I)$.

 Title Hilbert’s Nullstellensatz Canonical name HilbertsNullstellensatz Date of creation 2013-03-22 13:03:59 Last modified on 2013-03-22 13:03:59 Owner rmilson (146) Last modified by rmilson (146) Numerical id 8 Author rmilson (146) Entry type Theorem Classification msc 13A10 Synonym Nullstellensatz Related topic RadicalOfAnIdeal Related topic AlgebraicSetsAndPolynomialIdeals Defines zero set Defines Hilbert’s Nullstellensatz Defines weak Nullstellensatz