 Finite-field theorem for finitely generated integer algebras

A field that is a finite-type integer algebra has prime characteristic or zero. Prime characteristic and Zariski lemma make it a finite algebraic extension of a finite prime field. In characteristic zero, the same lemma makes it a number field; clearing finitely many coefficient denominators makes it integral over $\mathbb{Z}[1/N]$. The integral field extension forces the base domain to be a field, but a prime not dividing $N$ is not invertible there. This contradiction excludes characteristic zero.