A slick proof that a field which is finitely generated as a ring is finite

One approach is via the theory of Hilbert-Jacobson rings. There are several equivalent definitions, including that every prime ideal be the intersection of the maximal ideals containing it. From this it is easy to see that a PID, for instance, is a Hilbert-Jacobson ring iff it has infinitely many maximal ideals, and that in particular $\mathbb{Z}$ is a Hilbert-Jacobson ring.

Now here is an important and useful result about Hilbert-Jacobson rings:

Theorem: Let $R$ be a Hilbert-Jacobson ring, and $S$ a finitely generated $R$-algebra. Then:
a) $S$ is a Hilbert-Jacobson ring.
b) For every maximal ideal $\mathfrak{P}$ of $S$, $\mathfrak{p} := \mathfrak{P} \cap R$ is a maximal ideal of $R$.
c) The degree $[S/\mathfrak{P} : R/\mathfrak{p}]$ is finite.

(This result and its proof can be found in these notes: see Theorem 283 in $\S 12.2$.)

In particular every field which is a quotient of $\mathbb{Z}[t_1,\ldots,t_n]$ has finite degree over $\mathbb{Z}/(p)$ so is finite.

The simplest proof I know is the one here.