Is $\mathbb{Z}_p$ a Finite Field?

Question

Denote the integers modulo $p$, $\mathbb{Z}$ mod $P$, as $\mathbb{Z}_P$. Denote the set of integers equivalent to $n$ mod $P$ - the equivalence class of $n$ as $\overline{n}$.

We know that for any prime $p$, $\mathbb{Z}_P$ is a field. As a finite field contains a finite number of elements, and $\mathbb{Z}_P$ has elements $\overline{0}, \overline{1}, \ldots, \overline{p-1}$, which is obviously a finite set. So is $\mathbb{Z}_P$ a finite field?

(Could it be the case that $\mathbb{Z}_P$ is not a finite field, because though $\mathbb{Z}_P$ contains only $p$ elements, every one of these elements is an infinite set?)

Answer 1

Yes, it is a finite field. It doesn't matter that the elements of that field are themselves (infinite) sets, the "element of" relation is not transitive [in general, there are sets with the property that $x\in y \in T \Rightarrow x\in T$, but this is not one of them].

Answer 2

For any ring $R$ and ideal $I \leqslant R$, $R/I$ is a field if and only if $I$ is maximal. Let $R = \Bbb{Z}$ in your problem. Show that every prime ideal is maximal in $\Bbb{Z}$.