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".
Best Answer
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.