Let $\alpha$ be a root of $p(x)\in \mathbb F_{p}[x]$, irreducible of degree $n$. Show that $\alpha, \alpha^{p}, ..., \alpha^{p^{n-1}}$ are distinct roots of $p(x)$.
I can use the property that $a^{p}=a,a\in \mathbb{F}_{p}$ and $(a+b)^{p}=a^{p}+b^{p},a,b\in \mathbb{F}_{p}$ to prove that $\alpha, \alpha^{p}, ..., \alpha^{p^{n-1}}$ are indeed the roots of $p(x)$. But I have a hard time figuring out that these roots are distinct.
Observe that any irreducible polynomial in $F_p[X]$ must be separable. So the roots of $p(x)$ are all distinct. We have $F_p[\alpha]:F_p=n$. Suppose $\alpha^{p^i}=\alpha^{p^j}$ then we have $\alpha^{p^k}=\alpha$ for some $k<n$ and so $\alpha$ belongs to the splitting field of $X^{p^k}-X$ over $F_p$ .Then we have $F_p[\alpha]:F_p≤k<n$ a contradiction.