Suppose that $F$ is a field of $p^n$ elements.
For $a\in F, $ tr (a):= $a+a^p+ a^{p^2}+...+a^{p^{n-1}}$. Then, it is to be shown that tr(a) is in $F_p$ (finite field of order $p$).
Since $x^{p^n}-x=\Pi_{a\in F} (x-a)$, it suffices to show that tr$(a)^p=$ tr(a).
For this to be true, we must have $(a+a^p+ a^{p^2}+...+a^{p^{n-1}})^p= a^p+...+a= $tr$(a)$.
But I don't understand why that should be true as I know that if $F$ were of characteristic $p$, then we would have $(x+y)^p= x^p+y^p$. But in the above case, $F$ is not given to be of characteristic $0$.