From looking through the Wikipedia page about Cyclotomic Polynomials, I have seen a few of the properties but one that strikes me in particular is the fact that for distinct primes $p,q$, the $pq$-th cyclotomic polynomial $\Phi_{pq}(x)$ only have coefficients $-1,0,+1$
I'm interested in seeing how one might go about proving this, but haven't been able to find anything that made sense to me online.
One thought I had is using another property that I'd found of cyclotomic polynomials: namely that for coprime integers $n,m$:
$\Phi_n(t^m) = \Phi_n(t)\Phi_{nm}(t)$, giving us in this case the two equations:
$\Phi_q(t^p) = \Phi_q(t)\Phi_{pq}(t)$
$\Phi_p(t^q) = \Phi_p(t)\Phi_{pq}(t)$
Taking just the first one for now, we see that: $$\frac{\Phi_q(t^p)}{\Phi_q(t)} = \Phi_{pq}(t)$$
$$\Rightarrow (\frac{t^{pq}-1}{t^q-1})(\frac{t-1}{t^p-1}) = \Phi_{pq}(t)$$
$$\Rightarrow (\frac{(t^q-1)\Phi_p(t^q)}{t^q-1})(\frac{t-1}{t^p-1}) = \Phi_{pq}(t)$$
But at this point I can see that I'm really just about to get back to where I started.
Are there any other properties of cyclotomic polynomials that I might be able to exploit to derive the desired result?
Thank you.