Let $k$ be a field containing $\mathbb{F}_{p}$ and let $\alpha$ be a root in $k$ for $x^{p}-x-1 \in \mathbb{F}_{p}[x]$ Show that the roots of $x^{p}-x-1$ in $k$ are $\alpha, \alpha-1, \cdots, \alpha-p+1$
The question gives a hint of using $f(x-\alpha)$ with $f(x)=x^{p}-x$
In $\mathbb{F}_{p}[x]$ we have that $x^{p}-x=x(x-1) \cdots (x-(p-1))$ so we have roots $0,1, \cdots, p-1$ but I'm not sure how to use this fact to help in the above problem. I would appreciate a hint to get me started. Thanks.
Since the characteristic of $k$ is $p$, we have that $(\alpha - i)^p = \alpha^p - i^p = \alpha^p - i$, for any integer $i$.
Now plug $\alpha - i$ into the original polynomial $x^p - x - 1$.