Any Finite Field $K$ with $p^r$ element is a splitting field for $f(t) = t^{p^r} - t$

Question

Any Finite Field $K$ with $p^r$ element is a splitting field for $f(t) = t^{p^r} - t$.

I'm having some trouble convincing myself this is truth, if we factor out $t$, then then we equivalently saying every element $\alpha \in K$, $\alpha ^{p^r-1} = 1$.

Why is this the case? I done some searching on the net but could not find a explanation I understand.

Any help is appreciated.

Answer 1

The nonzero elements of $K$ form a group of order $p^r-1$ under multiplication. In a group of order $n$, each element satisfies $x^n=e$, the identity. So each nonzero element of $K$ satisfies $\alpha^{p^r-1}=1$.