Möbius Inversion Formula for Cyclotomic Polynomials

Corollary to Product of Cyclotomic Polynomials
Let $n > 0$ be a (strictly) positive integer.

Let $\Phi_n$ be the $n$th cyclotomic polynomial

Then:
 * $\displaystyle \Phi_n(x)=\prod_{d\mathop \backslash n}\left(x^d-1\right)^{\mu(d)}$

where:
 * the product runs over all divisors of $n$.
 * $\mu$ is the Möbius function.

Proof
By Product of Cyclotomic Polynomials,


 * $\displaystyle \prod_{d \mathop \backslash n} \Phi_d \left({x}\right) = x^n-1$

for all $n\in\mathbb N$.

The nonzero rational forms form an abelian group under multiplication.

By the Möbius inversion formula for abelian groups, this implies


 * $\displaystyle \Phi_n(x)=\prod_{d\mathop \backslash n}\left(x^d-1\right)^{\mu(d)}$

for all $n\in\mathbb N$.