A Lemma stated:
Let $F$ be a finite field, of prime characteristic $p$, and with algebraic closure $\overline{F}$. The polynomial $x^{p^{n}} - x$ has $p^{n}$ distinct zeros in $\overline{F}$.
The first line of the proof goes like this:
Since $\overline{F}$ is algebraically closed, $x^{p^{n}} - x$ factors into $p^{n}$ linear factors. So all that is left to show is that each factor does not appear more that once.
My question is how do we know that $\overline{F}$ is closed?