Weak Hilbert Nullstellensatz

For an algebraically closed field $k$, every maximal ideal of $k[x_1,\dots ,x_n]$ has the formfor a unique point $a\in k^n$. Equivalently, every proper ideal has a common zero.

If a field $L$ is finitely generated as an algebra over a subfield $k$, then $L$ is a finite algebraic extension of $k$. Applying this to a residue field of a polynomial ring proves the Weak Hilbert Nullstellensatz.