Hilbert's Nullstellensatz (strong)

Over an algebraically closed field, the ideal of a variety is the radical of the defining ideal.

Hilbert’s Nullstellensatz (strong)

Hilbert’s Nullstellensatz (strong): Let $k$ be an algebraically closed field and let $I\triangleleft k[x_1,\dots,x_n]$ be an ideal in the polynomial ring . Let

I(V(I))=\{f\in k[x_1,\dots,x_n] : f(a)=0\ \text{for all }a\in V(I)\}.

Then

I(V(I))=\sqrt{I},

where $\sqrt{I}$ denotes the radical of an ideal .

This identifies geometric vanishing with algebraic nilpotence modulo $I$ and implies, for instance, that varieties correspond to radical ideals and irreducible varieties correspond to prime ideals .