Hilbert's Nullstellensatz

Hilbert's Nullstellensatz states that if is an algebraically closed field and is an ideal of the polynomial ring , then the set of points satisfying for every determines the ideal radical of by

Here is the ideal of all polynomials that vanish on . Equivalently, if a polynomial vanishes at every common zero of , then for some positive integer . The weak Nullstellensatz says that every proper ideal of has a common zero in . The strong Nullstellensatz follows from the weak form by the Rabinowitsch trick.