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

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Thus the polynomials are equal.

$\blacksquare$

Also see

• Möbius Inversion Formula for Cyclotomic Polynomials