I need help proving that the polynomial $f(x)=(x-\alpha)(x-\alpha^p)\cdots(x-\alpha^{p^{n-1}})$ is in $F_p[x]$ if $\alpha\in F_{p^n}$.
This assertion is trivial for $\alpha$ already in $F_p$. So we assume it's not. We know this polynomial is degree $n$ and we know all of its roots are contained in $F_{p^n}$. We also have the result that $x^{p^n}-x$ is the product $\prod_{\alpha\in F_{p^n}}(x-\alpha)$. Thus we know that if all the $\alpha^{p^i}$ are distinct then $f(x)|x^{p^n}-x$.
My rough thoughts so far: We have a polynomial $f(x)$ of degree $n$ and we know its roots are all the powers of some element $\alpha$. Thus when we adjoin the root $\alpha$ to $F_p$ we have that the polynomial will split completely. I'm not sure where to go from here though.