Let $F$ be a finite field of characteristic $p$. Show that every element of $F$ is algebraic over $\mathbb{Z}_p$.
Since char$(F)$ = $p$, $\forall a \in F $ not zero, $ap$ = 0. We also have the same characteristic for $\mathbb{Z}_p$.
How can I use the prime characteristic to show there is a polynomial $P(x) \in \mathbb{Z}_p[x]$ such that $P(a) = 0$?