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 nth cyclotomic polynomial &Phi;'n(z) is defined by the equation

\Phi_n(z) = \prod_{(j,n)=1} (z - \zeta)^j \, $$

where 1 &le; j &le; n, only those j which are relatively prime to n are taken into the product, and &zeta; is a primitive n'th root of unity. For n > 2, &Phi;n(z) does not have any real roots. But we can always express zn &minus;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)