Strong Hilbert Nullstellensatz

For an ideal $I\subseteq k[x_1,\dots ,x_n]$ over an algebraically closed field,

Let $I,J\subseteq \mathbb{Z}[x_1,\dots ,x_n]$. If $V_{\mathbb{C}}(I)\subseteq V_{\mathbb{C}}(J)$, thenfor all but finitely many primes $p$. The Nullstellensatz puts a power of each generator of $J$ in $I\mathbb{Q}[x_1,\dots ,x_n]$; clearing denominators leaves only finitely many exceptional characteristics.

To prove the Strong Hilbert Nullstellensatz from its weak form, adjoin a variable $y$ and the equation $1-yf$. If $f$ vanishes on $V(I)$, the ideal $(I,1-yf)$ has no common zero and is therefore the unit ideal. Substituting $y=f^{-1}$ and clearing denominators yields $f^N\in I$ for some $N$.