Product of Cyclotomic Polynomials

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

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

where:
 * $\map {\Phi_d} x$ denotes the $d$th cyclotomic polynomial
 * the product runs over all divisors of $n$.

Proof
From the Polynomial Factor Theorem and Complex Roots of Unity in Exponential Form:
 * $\ds x^n - 1 = \prod_\zeta \paren {x - \zeta}$

where the product runs over all complex $n$th roots of unity.

In the, each factor $x - \zeta$ appears exactly once, in the factorization of $\map {\Phi_d} x$ where $d$ is the order of $\zeta$.

Thus the polynomials are equal.

Also see

 * Möbius Inversion Formula for Cyclotomic Polynomials