The discriminant of cyclotomic polynomial $\Phi_n(x)$

Question

Let $\Phi_n(x)$ be a cyclotomic polynomial, $n\in \mathbb{N}$. Prove that $\mathrm{Disc}(\Phi_n(x)) = (-1)^{\frac {\phi(n)}{2}}n^{\phi(n)}\prod_{p\mid n,\, p\text{ prime}} {p}^{\frac {\phi(n)}{1-p}}$

I've found a solution for prime $n$. For arbitrary $n$ we have $D(\Phi_n(x))=\prod_{\substack{1\le j<k\le n\\\gcd(j,n)=\gcd(k,n)=1}}(e^{\frac{2k\pi i}{n}}-e^{\frac{2j\pi i}{n}})^2$. Also we have $\Phi_n(X) = \frac{x^n-1}{\prod_{d|n, d<n} \Phi_d(x)}$. What can I get from that?

Answer 1

The computation can be reduced to prime powers, using the multiplicativity of Euler's totient function and "exploiting the fact that cyclotomic fields of relatively prime order are linearly disjoint", see Theorem $2$ in these notes, which gives a full proof. For prime powers it works like in the question you have linked; see also Proposition $4$ in the lecture notes above.