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Hilbert's Nullstellensatz


Let K be an algebraically closed field and let I be an ideal in K(x), where x=(x_1,x_2,...,x_n) is a finite set of indeterminates. Let p in K(x) be such that for any (c_1,...,c_n) in K^n, if every element of I vanishes when evaluated if we set each (x_i=c_i), then p also vanishes. Then p^j lies in I for some j. Colloquially, the theory of algebraically closed fields is a complete model.


See also

Algebraic Set, Ideal

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References

Becker, T. and Weispfenning, V. "The Hilbert Nullstellensatz." §7.4 in Gröbner Bases: A Computational Approach to Commutative Algebra. New York: Springer-Verlag, pp. 312-323, 1993.Hartshorne, R. Algebraic Geometry. New York: Springer-Verlag, 1977.

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Hilbert's Nullstellensatz

Cite this as:

Weisstein, Eric W. "Hilbert's Nullstellensatz." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/HilbertsNullstellensatz.html

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