Proving Mobius formula for cyclotomic polynomials without using $x^{n} - 1 = \prod_{d \mid n} \Phi_d(x)$

Question

I know that using the equation

$$x^{n} - 1 = \prod_{d \mid n} \Phi_d(x)$$

we can prove

$$\Phi_n(x) = \prod_{d\mid n} (x^d-1)^{\mu(n/d)}$$

by taking logs and using Mobius inversion. What if I don't want to use the first equation? Are there other methods to prove the second equation??

Answer 1

For instance, the inclusion-exclusion principle. $\Phi_n(x)$ is the minimal polynomial of a primitive $n$-th root of unity: we may label any $n$-th root of unity according to the degree of its minimal polynomial and recall that the roots of $x^d-1$ are the $d$-th roots of unity (primitive or not). By the inclusion-exclusion principle, $$ \prod_{d\mid n}(x^d-1)^{\mu(n/d)} $$ is the product of the terms $(x-\xi)$ where $\xi$ is a primitive $n$-th root of unity, i.e. $\Phi_n(x)$.