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,... , 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": 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. 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
