Hilbert's Nullstellensatz

A collection of related theorems linking ideals in polynomial rings over an algebraically closed field k to algebraic sets in affine space: classically, the (strong) Nullstellensatz states that for an ideal I ⊂ k[x1,…,xn], the ideal of functions vanishing on the zero set V(I) equals the radical rad(I); the weak form identifies maximal ideals with k-rational points.