Hilbert’s Nullstellensatz

Let K be an algebraically closed field, and let I be an ideal in K[x1,,xn], the polynomial ring in n indeterminates.

Define V(I), the zero setMathworldPlanetmathPlanetmath of I, by

V(I)={(a1,,an)Knf(a1,,an)=0 for all fI}

Weak Nullstellensatz:
If V(I)=, then I=K[x1,,xn]. In other words, the zero set of any proper idealMathworldPlanetmath of K[x1,,xn] is nonempty.

Hilbert’s (Strong) Nullstellensatz:
Suppose fK[x1,,xn] satisfies f(a1,,an)=0 for every (a1,,an)V(I). Then frI for some integer r>0.

In the of algebraic geometryMathworldPlanetmathPlanetmath, the latter result is equivalent to the statement that Rad(I)=I(V(I)), that is, the radicalPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath 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