Question in proof of the existence of a field of order $p^n$

Question

I'm trying to understand a detail in the proof that there exists a finite field of order $p^n$. (Here $p \in \mathbb{Z}^{+}$ is a prime and $n \in \mathbb{N}$.) I've found questions on the proof of this theorem posted here already, which have been helpful (e.g. this one), but there's a particular detail that is bugging me and for which I haven't found an explanation.

What I understand so far is that the polynomial $X^{p^n} - X \in \mathbb{Z}_p[X]$ has an extension $E[X]$ which contains $p^n$ distinct roots in $E$. Let $c_1,c_2,\ldots,c_{p^n} \in E$ be the distinct roots. Now we want to show that $K = \{c_1,c_2,\ldots,c_{p^n}\}$ is a field, so we need to show that $K$ is closed under addition (among other things). All the proofs I've seen for this seem to implicitly assume that $a^{p^n} = a$ for any (nonzero) $a \in K$, without any explanation. My question: how can we justify this?

I have learned that if $F$ is a finite field with $q$ elements, then $a^{q} = a$ for all $a \neq 0$ in $F$. But we can't use this property here because we haven't yet established that $K$ is a field (because that's what we're trying to show!). We know that $c_1,\ldots,c_{p^n}$ are in the bigger field $E$, but $E$ would have more than $p^n$ elements (we don't even know how many), so I don't see how Fermat's Little Theorem could help us here.

Could anyone shed some light on this? Thanks in advance.

Answer 1

This does not have much to do with finite fields: if $\alpha$ is a root of $X^n -X$ in the algebraic closure $\overline{k}$ of some field $k$, then

$$ \alpha^n - \alpha = 0 $$

and thus $\alpha^n = \alpha$ in $\overline{k}$. The operations are done in $\overline{k}$ where that equality makes perfect sense.

Answer 2

I see guidoar's answer but I'm not sure if it addresses your concern so I write another answer and hope that I don't misunderstand your question.

I believe what you want to know is why $K$ contains precisely $p^n$ elements (and no more). And among other things as you mentioned, you want to check that $K$ is closed under addition. Let $a\in K$ and $b\in K$. Since $K$ has characteristic $p$, $(a+b)^p=a^p+b^p$. By induction, $(a+b)^{p^n}=a^{p^n}+b^{p^n}=a+b$, hence the claim.