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

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 $n \in \N$.

The nonzero rational forms form an abelian group under multiplication.

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


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

for all $n \in \N$.