Motivation:
Say you have a noetherian scheme of finite type $X$ over $\mathbb Z$. Then to every closed point $x$ corresponds a norm, which is the number of elements in the residue field (which we know to be finite)
$$ N(x) = |k(x)|. $$
So, this number will be $p^n$ for some prime $p$ and integer $n$. We want to prove that the number
$$ a(p,k) = \{x \text{ closed point }\mid N(x) = p^k \} $$
is finite.
Given this, we reduced the problem to the following:
Let $X = \textrm{spec }\mathbb{F}_p[T_1, \dots, T_n] = \textrm{spec }R$ defined over $\mathbb F_p$. The closed points now correspond to maximal ideals of $R$, and therefore, we want to prove that for each $k$ there are finitely many ideals $\mathfrak m$ such that $R/\mathfrak{m} = p^k$.
If $n=1$ this is easy to determine (since $R$ will be a PID, we can count irreducible polymonials of given degree). For a generic $n$ we have no clue how to proceed.
One strategy may be to use the Nullstellensatz on $\overline{\mathbb{F}_p}$ and hope we can get something back by using the Frobenius morphism.