Show that if $x^p-x-1$ is reducible in $F[x]$ where $F$ has characteristic $p$ then it splits in $F[x]$ into monic distinct factors. So far, I have shown that the function is separable, after all, $D_f(x)=-1$ so they are relatively prime. I am having trouble showing that it splits. Our field $F$ contains $GF(F_p)$ thus it contains the splitting field of $x^n-1$. However, i am not sure how helpful that is.

Hint: $x$ is a root means that $x^p = x+1$; but $(x+1)^p = x^p+1$, so $x+1$ is a root too. 
