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 $n \in \N$.

$\blacksquare$