TITLE: Value of cyclotomic polynomial evaluated at 1
QUESTION [13 upvotes]: Let $\Phi_n(x)$ be the usual cyclotomic polynomial (minimal polynomial over the rationals for a primitive nth root of unity).
There are many well-known properties, such as $x^n-1 = \Pi_{d|n}\Phi_d(x)$.
The following fact appears to follow pretty easily:
Fact:
$\Phi_n(1)=p$ if $n$ is a prime power $p^k$.
$\Phi_n(1)=1$ if $n$ is divisible by more than one prime.
My question is, is there a reference for this fact? Or is it simple enough to just call it "folklore" or to just say it "follows easily from properties of cyclotomic polynomials".

REPLY [14 votes]: Another proof follows directly from the formula $X^{n} - 1 = \prod_{d \mid n} \Phi_d(x)$, since we can deduce from it that 
\begin{equation}
X^{n-1} + \cdots + X + 1 = \prod_{d \mid n, d>1} \Phi_d(x).
\end{equation} 
Thus, if $n = p^{k}$, we have 
$$
X^{p^{k}-1} + \cdots + X + 1 = \Phi_{p}(x) \cdots \Phi_{p^{k-1}}(x) \Phi_{p^{k}}(x).
$$
After evaluating in 1 we obtain $p^{k} = \Phi_{p}(1) \cdots \Phi_{p^{k-1}}(1) \Phi_{p^{k}}(1)$ and induction on $k$ gives $\Phi_{p^{k}}(1) = p$ for all $k$.
If $n = p_{1}^{\alpha_{1}} \cdots  p_{r}^{\alpha_{r}}$, where $\alpha_{i}$'s are positive integers and $r \geq 2$, then 
$$
n = \Phi_{n}(1) \prod_{d \mid n, d\neq 1,n} \Phi_d(1).
$$
If we assume the statement true for all positive integers $<n$ then the product in the left member of the equation equals $n$, since 
$$
\prod_{i=1}^{r}\Phi_{p_{i}}(1) \cdots \Phi_{p_{i}^{\alpha_{i}}}(1) = p_{1}^{\alpha_{1}} \cdots p_{r}^{\alpha_{r}} = n
$$
and the rest of the factors are 1. Thus, $\Phi_{n}(1) = 1$ also.