When I teach finite fields, one fun corollary is that $n \mid \phi(p^n-1)$, where $\phi$ is the Euler-phi function and $p$ is prime.
I spent several minutes in my office after class one day looking for an obvious reason. But I failed. I couldn't see a proof that doesn't essentially recast the argument that the cyclotomic polynomial $\Phi_{p^n-1}(x)$ splits into all primitive polynomials of degree $n$ in $\mathbb{Z}_p[x]$. Is there an elementary (let's say "polynomial-free") proof of this divisibility?
More generally: Prove that for all positive integers $a,n$ with $a > 1, n | \phi(a^{n}-1)$.
