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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?

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    The definition I had while asking this question (3 years ago), was different than the one most people use. The book I was using supplied a more direct concreted definition using roots of polynomials, but then proceeded to work with the abstract one instead. I found this confusing, because although I could clearly see that $\overline{F}$ *ought* to be algebraically closed, it was not yet established.2015-03-08

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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.)

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    The definition used at the time was a different ( but equivalent one ). I agree the one you mention is more natural and now the one I think of [if ever I find myself in the world of algebra :) ]. At the time of asking I didn't know they were equivalent.2014-02-19