Irreducible polynomial over field of order p Let $p$ be a prime and $F=\mathbb{Z}/p\mathbb{Z}$ and $f(t)\in F[t]$ be an irreducible polynomial of degree $d$.
I need to show that $f(t)$ divides $t^{(p^{n})}-t$ if and only if $d$ divides $n$.

The first theorem of the following paper provides the proof:
http://www.jstor.org/stable/2316211
Or look at Theorem $7.6$ of the following article 
http://www-groups.mcs.st-and.ac.uk/~neunhoef/Teaching/ff/ffchap3.pdf
