Talk:Cyclotomic identity

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 nth 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 nth 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)