Hilbert’s Nullstellensatz
Let be an algebraically closed field, and let be an ideal in , the polynomial ring in indeterminates.
Define , the zero set of , by
Weak Nullstellensatz:
If , then . In other words, the zero set of any proper ideal of is nonempty.
Hilbert’s (Strong) Nullstellensatz:
Suppose satisfies for every . Then for some integer .
In the of algebraic geometry, the latter result is equivalent to the statement that , that is, the radical of is equal to the ideal of .
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 |