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Concept
Hilbert's Nullstellensatz
Summary
In mathematics, Hilbert's Nullstellensatz (German for "theorem of zeros", or more literally, "zero-locus-theorem") is a theorem that establishes a fundamental relationship between geometry and algebra. This relationship is the basis of algebraic geometry. It relates algebraic sets to ideals in polynomial rings over algebraically closed fields. This relationship was discovered by David Hilbert, who proved the Nullstellensatz in his second major paper on invariant theory in 1893 (following his seminal 1890 paper in which he proved Hilbert's basis theorem).
Let k be a field (such as the rational numbers) and K be an algebraically closed field extension (such as the complex numbers). Consider the polynomial ring and let I be an ideal in this ring. The algebraic set V(I) defined by this ideal consists of all n-tuples x = (x1,...,xn) in Kn such that f(x) = 0 for all f in I. Hilbert's Nullstellensatz states that if p is some polynomial in that vanishes on the algebraic set V(I), i.e. p(x) = 0 for all x in V(I), then there exists a natural number r such that pr is in I.
An immediate corollary is the weak Nullstellensatz: The ideal contains 1 if and only if the polynomials in I do not have any common zeros in Kn. It may also be formulated as follows: if I is a proper ideal in then V(I) cannot be empty, i.e. there exists a common zero for all the polynomials in the ideal in every algebraically closed extension of k. This is the reason for the name of the theorem, which can be proved easily from the 'weak' form using the Rabinowitsch trick. The assumption of considering common zeros in an algebraically closed field is essential here; for example, the elements of the proper ideal (X2 + 1) in do not have a common zero in
With the notation common in algebraic geometry, the Nullstellensatz can also be formulated as
for every ideal J. Here, denotes the radical of J and I(U) is the ideal of all polynomials that vanish on the set U.
Official source
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