Hilbert's Nullstellensatz

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Hilbert's Nullstellensatz (German: "theorem of zeros," or more literally, "zero-locus-theorem") is a theorem which makes precise a fundamental relationship between the geometric and algebraic sides of algebraic geometry, an important branch of mathematics. It relates algebraic sets to ideals in polynomial rings over algebraically closed fields. The theorem was first proved by David Hilbert, after whom it is named.



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 k[X1,X2,..., Xn] and let I be an ideal in this ring. The affine variety 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 k[X1,X2,..., Xn] which vanishes on the variety 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 I in k[X1,X2,..., Xn] contains 1 if and only if the polynomials in I do not have any common zeros in Kn.

When k=K the "weak Nullstellensatz" may also be stated as follows: if I is a proper ideal in K[X1,X2,..., Xn], then V(I) cannot be empty, i.e. there exists a common zero for all the polynomials in the ideal. 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 that K be algebraically closed is essential here; the elements of the proper ideal (X2 + 1) in R[X] do not have a common zero. The weak Nullstellensatz is a generalization of the fundamental theorem of algebra, which is the case n = 1, I in this case being the proper ideal generated by the polynomial, and V(I) being the zeros.

With the notation common in algebraic geometry, the Nullstellensatz can also be formulated as

for every ideal J. Here, \sqrt{J} denotes the radical of J and I(U) is the ideal of all polynomials which vanish on the set U.

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