Hilbert’s Nullstellensatz

Let $K$ be an algebraically closed field, and let $I$ be an ideal in $K[x_1,\mathrm{\dots },x_n]$, the polynomial ring in $n$ indeterminates.

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

$$V(I)=\{(a_1,\mathrm{\dots },a_n)\in K^n\mid f(a_1,\mathrm{\dots },a_n)=0\text{ for all }f\in I\}$$

Weak Nullstellensatz:
If $V(I)=\mathrm{\varnothing }$, then $I=K[x_1,\mathrm{\dots },x_n]$. In other words, the zero set of any proper ideal of $K[x_1,\mathrm{\dots },x_n]$ is nonempty.

Hilbert’s (Strong) Nullstellensatz:
Suppose $f\in K[x_1,\mathrm{\dots },x_n]$ satisfies $f(a_1,\mathrm{\dots },a_n)=0$ for every $(a_1,\mathrm{\dots },a_n)\in V(I)$. Then $f^r\in I$ for some integer $r>0$.

In the of algebraic geometry, the latter result is equivalent to the statement that $\mathrm{Rad}(I)=I(V(I))$, 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