Product of Cyclotomic Polynomials

Theorem

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

Then:

$\displaystyle \prod_{d \mathop \backslash n} \Phi_d \left({x}\right) = x^n-1$

where:

$\Phi_d \left({x}\right)$ denotes the $d$th cyclotomic polynomial
the product runs over all divisors of $n$.

Proof

$\displaystyle x^n - 1 = \prod_\zeta \left({x - \zeta}\right)$

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

In the left hand side, each factor $x - \zeta$ appears exactly once, in the factorization of $\Phi_d \left({x}\right)$ where $d$ is the order of $\zeta$.

Thus the polynomials are equal.

$\blacksquare$