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?
Best Answer
The comments already stated as much, but I'm posting an answer to get this question out of the unanswered queue.
The algebraic closure of any field is algebraically closed by definition. Being algebraically closed is the key defining property of the algebraic closure. Details depend on what definition you use, but defining it as an algebraic field extension which is algebraically closed should be quite common and makes this property clear.
(Note however that this algebraic closure will no longer be finite.)