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

Then

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 .

Proof sketch: The inclusion $\sqrt{I}\subseteq I(V(I))$ is immediate. For the reverse inclusion one uses the Rabinowitsch trick: if $f\notin \sqrt{I}$, then $I+(1-tf)\subset k[x_1,\dots,x_n,t]$ is a proper ideal, hence has a common zero by the weak form, which forces a point in $V(I)$ where $f$ does not vanish.