Let p be prime. For $n|p-1$, let $f_n(x)=(x-a_1)(x-a_2)\cdots(x-a_s)$ where $a_1$ through $a_s$ belong to the exponent $n \pmod{p}$ and $s=\phi(n)$. For sufficiently large x, $f_n(x)$ is positive. Prove that $f_n(x)=\prod_{m|n} (x^{n/m} - 1)^{\mu(m)} \pmod{p}$
I did prove (using logs) that if we let $f(n)$ be a positive function over the natural numbers and let $g(n) = \prod_{m|n} f(m)$,
then $f(n)=\prod_{m|n} g(n/m)^{\mu(m)}.$
So, if I can prove $x^n-1=\prod_{m|n} f_m(x) \pmod{p}$, then I can use what I have already proven to finish the proof. I'm not sure where to go from here, though.