Zariski's Lemma is the following:



> 
> Let $K$ be a field and $R$ be a $K$-algebra with $R=K[x\_1,\dots,x\_n]$
> for some $x\_1,\dots,x\_n\in R$. If $R$ is a field then $x\_1,\dots,x\_n$
> are algebraic over $K$.
> 
> 
> 


Oskar Zariski used this Lemma to prove Hilbert's Nullstellensatz. Is there another non-trivial application of this Lemma?



Application of Zariski's Lemma other than Hilbert's Nullstellensatz
You can use Zariski's lemma to show that if $R$ is a finitely generated $\mathbf Z$-algebra, then its residue fields $R/\mathfrak m$ for maximal ideals $\mathfrak m$ are all finite. That leads to a description of all the maximal ideals in $\mathbf Z[x]$, for instance. See Section 5 [here](https://kconrad.math.uconn.edu/blurbs/ringtheory/maxideal-polyring.pdf).


