numtheory/cyclotomic(deprecated) - Help

numtheory

 cyclotomic
 calculate cyclotomic polynomial

 Calling Sequence cyclotomic(n, t)

Parameters

 n - non-negative integer t - variable

Description

 • Important: The numtheory[cyclotomic] command 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) =

$\prod _{\mathrm{\zeta }}\left(t-\mathrm{\zeta }\right):\mathrm{\zeta }\mathrm{is}a\mathrm{primitive}n\mathrm{th}\mathrm{root}\mathrm{of}\mathrm{unity}$

 • 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 $\mathrm{cyclotomic}\left(n,t\right)=\frac{{t}^{n}-1}{\prod _{d=\mathrm{divisors}\left(n\right)∖\left\{n\right\}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\mathrm{cyclotomic}\left(d,t\right)}$.
 • The degree of the nth cyclotomic polynomial is given by the totient function numtheory[phi].

Examples

 > $\mathrm{with}\left(\mathrm{numtheory}\right):$
 > $\mathrm{cyclotomic}\left(1,x\right)$
 ${x}{-}{1}$ (1)
 > $\mathrm{cyclotomic}\left(20,z\right)$
 ${{z}}^{{8}}{-}{{z}}^{{6}}{+}{{z}}^{{4}}{-}{{z}}^{{2}}{+}{1}$ (2)