Möbius Inversion Formula for Cyclotomic Polynomials

Theorem
Let $n > 0$ be a (strictly) positive integer.

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

Then:
 * $\map {\Phi_n} x = \ds \prod_{d \mathop \divides n} \paren {x^d - 1}^{\map \mu {n / d} }$

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

Proof
By Product of Cyclotomic Polynomials:


 * $\ds \prod_{d \mathop \divides n} \map {\Phi_d} x = x^n - 1$

for all $n \in \N$.

The nonzero rational forms form an abelian group under multiplication.

By the Möbius inversion formula for abelian groups, this implies:


 * $\ds \map {\Phi_n} x = \prod_{d \mathop \divides n} \paren {x^d - 1}^{\map \mu {n / d} }$

for all $n \in \N$.