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单词 HilbertsNullstellensatz
释义

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).

TitleHilbert’s Nullstellensatz
Canonical nameHilbertsNullstellensatz
Date of creation2013-03-22 13:03:59
Last modified on2013-03-22 13:03:59
Ownerrmilson (146)
Last modified byrmilson (146)
Numerical id8
Authorrmilson (146)
Entry typeTheorem
Classificationmsc 13A10
SynonymNullstellensatz
Related topicRadicalOfAnIdeal
Related topicAlgebraicSetsAndPolynomialIdeals
Defineszero set
DefinesHilbert’s Nullstellensatz
Definesweak Nullstellensatz
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