Talk:Cyclotomic identity

Definition of the cyclotomic polynomials

This article contained the assertion that $z^{n}-1$ is a cyclotomic polynomial. I have changed the article to say that $z^{n}-1$ is the product of cyclotomic polynomials. Here's the explanation.

The n'th cyclotomic polynomial Φ_n(z) is defined by the equation

$\Phi _{n}(z)=\prod _{(j,n)=1}(z-\zeta )^{j}\,$

where 1 ≤ j ≤ n, only those j which are relatively prime to n are taken into the product, and ζ is a primitive n'th root of unity. For n > 2, Φ_n(z) does not have any real roots. But we can always express zⁿ −1 as the product of cyclotomic polynomials:

$z^{n}-1=\prod _{d|n}\Phi _{d}(z),\,$

where the product runs over the divisors of n. DavidCBryant 00:33, 18 May 2007 (UTC)