= Solution
Let $K$ be a field finitely generated as a $\mathbb Z$-algebra. If $K$ has positive <characteristic> $p$, it is a finitely generated algebra over $\mathbb F_p$; <Zariski lemma> makes it a finite algebraic extension of the finite field $\mathbb F_p$, so $K$ is finite.
Suppose instead that $K$ has characteristic zero. Then $K$ is a finitely generated $\mathbb Q$-algebra and Zariski lemma makes it a <number field>. For algebra generators $\alpha_1,\ldots,\alpha_r$, choose a nonzero integer $N$ such that every $\alpha_i$ is integral over $\mathbb Z[1/N]$. The whole algebra $K=\mathbb Z[\alpha_1,\ldots,\alpha_r]$ would then be integral over $\mathbb Z[1/N]$. But for a prime $q\nmid N$, the element $1/q\in K$ is not integral over the integrally closed domain $\mathbb Z[1/N]$, a contradiction. Hence every field finitely generated over the integers is finite, and in particular no infinite field has that property.
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