Let $K$ be a field, which is a finitely generated $\mathbb{Z}$-algebra.
Show that $K$ has characteristic $> 0$.
I have this hint but I am not quit sure that I understand it; if $\operatorname{char}(K) = 0$, then $\mathbb{Z} \subset K$, then $\Bbb Q \subset K$,
and $K$ is a finitely generated $\Bbb Q$-algebra. Use Zariski's Lemma to show that $K$ is a
finitely generated $\Bbb Q$-module, and then Artin-Tate Lemma to get a contradiction
Best Answer
If $\operatorname{char}(K) = 0$, then $\mathbb{Z} \subset K$, then $\Bbb Q \subset K$, and $K$ is a finitely generated $\Bbb Q$-algebra. By Zariski's Lemma $K$ is a finitely generated $\Bbb Q$-module, and by Artin-Tate Lemma we get that $\mathbb Q$ is a finitely generated $\mathbb Z$-algebra, a contradiction.