 Zariski's lemma

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.

Zariski's lemma is a result in algebraic geometry that is named after the mathematician Oscar Zariski. It provides a condition for the vanishing of a polynomial function on an algebraic variety.