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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    $\begingroup$ Isn't that the definition? That $\overline{F}$ is the intersection of all algebraically closed fields containing $F$; and that the intersection of algebraically closed fields is algebraically closed? $\endgroup$ – Asaf Karagila Aug 3 '12 at 18:33
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    $\begingroup$ Or it's defined as an algebraic extension that is algebraically closed. $\endgroup$ – JSchlather Aug 3 '12 at 18:35
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    $\begingroup$ I was confusing it with the algebraic closure of $F$ in some extension field $E$. Since there is no such $E$ in this context, your definition must be the case. Clearly I should re-skim this chapter before going forward. Thanks! $\endgroup$ – roo Aug 3 '12 at 18:36
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    $\begingroup$ Check out Dummit and Foote, p.543, Prop 29 if that's what you're asking. $\endgroup$ – user12802 Aug 3 '12 at 19:14
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    $\begingroup$ By definition, an algebraic closure of a field $K$ is an algebraically closed extension $\overline{K}$ of $K$ which is algebraic over $K$. So it's a matter of definition. $\endgroup$ – Keenan Kidwell Aug 3 '12 at 20:40

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.)

  • $\begingroup$ 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. $\endgroup$ – roo Feb 19 '14 at 20:55

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