In p.35 of Miranda's book Algebraic Curves and Riemann Surfaces, the following theorem is stated and is called as Hilbert's Nullstellensatz.
Theorem. Let $f\in \Bbb C[x_1,\dots,x_n]$ is an irreducible polynomial, and $h$ is a polynomial vanishing everywhere on the zero set of $f$. Then $f$ divides $h$.
But as I remember, the original Nullstellensatz is the following:
Theorem. Every maximal ideal of $\Bbb C[x_1,\dots,x_n]$ is of the form $(x_1-a_1,\dots,x_n-a_n)$.
How does the original Nullstellensatz imply the new one?