Product of Cyclotomic Polynomials

Theorem
Let $n>0$ be a positive integer.

Then


 * $\displaystyle\prod_{d\mid n}\Phi_d(x)=x^n-1$

where $\Phi_d(x)$ denotes the $d$th cyclotomic polynomial and the product runs over all divisors of $n$.

Proof
From Polynomial Factor Theorem and Complex Roots of Unity in Exponential Form, we have
 * $x^n-1=\displaystyle\prod_\zeta(x-\zeta)$

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

In the LHS, each factor $x-\zeta$ appears exactly once, in the factorization of $\Phi_d(x)$ where $d$ is the order of $\zeta$. Thus the polynomials are equal.