Hilbert's Nullstellensatz

Hilbert's Nullstellensatz (German: "theorem of zeros") is a theorem in algebraic geometry that relates varieties and ideals in polynomial rings over algebraically closed fields. It was first proved by David Hilbert.

Let K be an algebraically closed field (such as the complex numbers), consider the polynomial ring K[X₁,X₂,... , X_n] 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,...,x_n) 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,... , X_n] 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 p^r is in I.

An immediate corollary is the "weak Nullstellensatz": if I is a proper ideal in K[X₁,X₂,... , X_n], 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. Note that the assumption that K be algebraically closed is essential here: the proper ideal (X² + 1) in R[X] does not have a common zero.

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

I(V(J)) = √J    for every ideal J

Here √J denotes the radical of J and I(\U) denotes the ideal of all polynomials which vanish on the set U. In this way, we obtain an order-reversing bijective correspondence between the affine varieties in Kⁿ and the radical ideals of K[X₁,X₂,... , X_n].