I'm trying to solve this question:
Let $I \subset \mathbb{Z}[x]$ be a maximal ideal. Show that if $\mathbb{Z}[x]/I$ has characteristic $p > 0$, then $\mathbb{Z}[x]/I$ is a finite field.
and I thought the forrowing proposition is useful for this:
If a finitely generated ring K is a field, it is a finite field. (Atiyah-Macdonald, exercise7-7)
I know if a field K has characteristic $p>0$, then K includes a primary field $\mathbb{Z}/p\mathbb{Z}$. But can I say then K is finitely generated as $\mathbb{Z}/p\mathbb{Z}$-algebra then? In this case, $\mathbb{Z}[x]/I$ is a noetherian ring, so I thought it is finitely generated as $\mathbb{Z}/p\mathbb{Z}$-algebra but I think it's different a little bit, noetherian rings and finitely generated algebra.. I'm confused.