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 $\zeta$.
 | This article, or a section of it, needs explaining. In particular: The above statement needs justification. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Explain}} from the code. |
Thus the polynomials are equal.
$\blacksquare$