Important: The numtheory package has been deprecated.  Use the superseding command NumberTheory[CyclotomicPolynomial] instead.

The function cyclotomic(n, t) returns the nth cyclotomic polynomial in t.

By definition, cyclotomic(n, t) =

With the exception of small values of n, for which a table lookup is used, $\mathrm{cyclotomic}\left(n\,t\right)$ is computed using an algorithm based on the fact that $\mathrm{cyclotomic}\left(n\,t\right)=\frac{t^n-1}{\prod _{d=\mathrm{divisors}\left(n\right)\setminus \left\{n\right\}}\mathrm{cyclotomic}\left(d\,t\right)}$.

The degree of the nth cyclotomic polynomial is given by the totient function numtheory[phi].

$\mathrm{with}\left(\mathrm{numtheory}\right)\:$

$\mathrm{cyclotomic}\left(1\,x\right)$

$x-1$

$\mathrm{cyclotomic}\left(20\,z\right)$

$z^8-z^6+z^4-z^2+1$