How to prove the polynomial $ x^{2011}-x\in \mathbb{Z}_{2011}[x] $ is separable?

Question

I am trying to prove that the polynomial $ x^{2011}-x\in \mathbb{Z}_{2011}[x] $ is separable.

Definition: Let K be a field and $f\in K[x]$ an irreducible polynomial. The polynomial $f$ is said to be separable if in some splitting field of $f$ over $K$ every root of $f$ is a simple root.

My attempt: $$\begin{align*} x^{2011}-x&=x(x^{2010}-1)\\ &=x(x^{1005}-1)(x^{1005}+1)\\ &=x(x-1)(x^{1004}+x^{1003}+\cdots+x^{2}+x+1)(x^{1005}+1) \end{align*}$$

But I have no idea what to do next. Any hints/ideas are much appreciated. Thanks in advance for any replies.

Answer 1

Hint: What does Fermat's little theorem say about the roots of that polynomial?

Answer 2

Another hint: A polynomial $f(x)$ has roots of multiplicities $>1$ if and only if $\gcd(f,f')$ has degree $>0$.