Dear MO_World,

I'm working on an ergodic theory question (about a generalization of eigenfunctions for measure-preserving transformations) and have run into a number theory question concerning cyclotomic polynomials that I'm unable to tackle.

The question is this:

Let $p$ be a prime and let $p|n$. When is it the case that $\Phi_n(e^{2\pi i/p})=\pm e^{2\pi ij/p}$ for some $j$?

Here $\Phi_n$ denotes the $n$th cyclotomic polynomial.

I've experimented with Mathematica and have found there are non-trivial cases in which the condition holds, whereas for most cases it does not seem to hold.

Letting $c(n,p)=\Phi_n(e^{2\pi i/p})$, we have $c(105,3)=1$, but $c(105,5)$ and $c(105,7)$ are not on the unit circle. None of $c(15,3)$, $c(21,3)$, $c(15,5)$, $c(35,5)$, $c(21,7)$, $c(35,7)$ are on the unit circle; $c(40,2)=1$, but $c(50,2)=5$...

Not surprisingly it seems to be easiest for the condition to hold for small $p$.

Also, using the relations $\Phi_{p^2n}(x)=\Phi_{pn}(x^p)$; and $\Phi_n(1)=q$ if $n=q^k$ for some prime $q$ and an integer $k$, but $\Phi_n(1)=1$ otherwise, it's not hard to see that the condition holds whenever $p^2|n$, but $n$ is not a power of $p$.

Thanks for any more systematic suggestions...