help with this interpretation..
Choose a prime $l$ such that $l\neq p$ and $l$ is a quadratic residue modulo $p$. Choose an integer $m\geq 1$ such that $l^m-1$ is divisible by $p$. Let $\theta$ be a primitive element of $F_{l^m}$ and pun $\alpha=\theta^{(l^m-1)/p}$. Then, the order of $\alpha$ is $p$; i.e., $1=\alpha^0=\alpha^p,\alpha^1=\alpha,\alpha^2,\alpha^3,...,\alpha^{p-1}$ are pairwise distinct and $x^p-1=\prod_{i=0}^{p-1}(x-\alpha^i).$
I know that $(\alpha)^p=(\theta^{(l^m-1)/p})^p=\theta^{l^m-1}=1$ but why $1=\alpha^0=\alpha^p,\alpha^1=\alpha,\alpha^2,\alpha^3,...,\alpha^{p-1}$ are pairwise distinct and $x^p-1=\prod_{i=0}^{p-1}(x-\alpha^i).$ ??