Show that $\mathbb{F}_q$ is a Splitting Field for Polynomial $x^q-x$ I have been given a homework problem which asks me to assume that $\mathbb{F}_q$ is a field of order $q$, and to show that $\mathbb{F}_q$ is a splitting field for the polynomial $x^{q}-x\in\mathbb{Z}_p[x]$, using Lagrange's theorem on the multiplicative group of units in $\mathbb{F}_q$.
Can you please help me to find a suitable approach to this? Thank you very much.
 A: Let $f(x) = x^q - x$.    To show that $f$ splits in $\mathbb{F}_q$, we wish to demonstrate the existence of $q$ roots of $f$ inside of this field, perhaps counting multiplicity.  Now if $x$ is a root of $f$, then notice it must satisfy $x^q = x$.  Trivially, this holds for $0$, so let's turn our focus to the invertible elements.
The multiplicative group $\mathbb{F}_q^\times$ has order $q-1$.  By Lagrange's theorem, the order of every element of $\mathbb{F}_q^\times$ must divide $q-1$.  In other words, for a given $x \in \mathbb{F}_q^\times$, we have $m|x| = q-1$ for some $m \in \mathbb{N}$.  Therefore, $x^{q-1} = x^{m|x|} = (x^{|x|})^m = 1^m = 1$.
Can you take it from here?

Relevance: 
This exercise shows that every finite extension of a finite field is a normal extension.  It is also true that such extensions are separable, from which we can conclude that all such extensions are Galois extensions, which are of great importance in Galois theory.
A: Hint: If $x$ is a non-zero element of $F_q$, then $x^{q-1}=1$. 
