# Is the Algebraic Closure of a Finite Field Algebraically Closed?

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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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? – Asaf Karagila Aug 3 '12 at 18:33
Or it's defined as an algebraic extension that is algebraically closed. – JSchlather Aug 3 '12 at 18:35
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! – roo Aug 3 '12 at 18:36
Check out Dummit and Foote, p.543, Prop 29 if that's what you're asking. – TheBirdistheWord Aug 3 '12 at 19:14
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. – Keenan Kidwell Aug 3 '12 at 20:40