The IsCyclotomicPolynomial function determines whether p is a cyclotomic polynomial, and, optionally, the order of p if p is cyclotomic.

The IsCyclotomicPolynomial(p, x) calling sequence returns true if p(x) is a cyclotomic polynomial, and false otherwise.

Use output_opt to specify whether to return the result, the order, or both:

output = result: Returns true if p is cyclotomic and false otherwise. This is the default behavior for IsCyclotomicPolynomial.

output = order: Returns the order of p as a cyclotomic polynomial if p is cyclotomic. Returns FAIL otherwise.

output = [result, order] (or output = [order, result]): Returns an expression sequence with result and then order (or with order and then result).

If p is the nth cyclotomic polynomial, then p is said to be the order of n.

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

$\mathrm{CyclotomicPolynomial}\left(2\,x\right)$

$x+1$

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

$\mathrm{true}$

$\mathrm{IsCyclotomicPolynomial}\left(x+2\,x\right)$

$\mathrm{false}$

$\mathrm{IsCyclotomicPolynomial}\left(x-\frac{1}{2}\,x\,\mathrm{output}=\mathrm{order}\right)$

$\mathrm{FAIL}$

$\mathrm{IsCyclotomicPolynomial}\left(\mathrm{CyclotomicPolynomial}\left(33\,x\right)\,x\,\mathrm{output}=\left[\mathrm{result}\,\mathrm{order}\right]\right)$

$\mathrm{true},33$

$p≔\mathrm{CyclotomicPolynomial}\left(7\,x\right)$

$p≔x^6+x^5+x^4+x^3+x^2+x+1$

$\mathrm{zeroes}≔\left[\mathrm{solve}\left(p=0\,x\right)\right]\:$

$q≔\mathrm{mul}\left(x-\mathrm{zeroes}\left[i\right]\,i=1..6\right)$

$q≔\left(x-\cos \left(\frac{2\mathrm{\pi }}{7}\right)-\mathrm{I}\sin \left(\frac{2\mathrm{\pi }}{7}\right)\right)\left(x+\cos \left(\frac{3\mathrm{\pi }}{7}\right)-\mathrm{I}\sin \left(\frac{3\mathrm{\pi }}{7}\right)\right)\left(x+\cos \left(\frac{\mathrm{\pi }}{7}\right)-\mathrm{I}\sin \left(\frac{\mathrm{\pi }}{7}\right)\right)\left(x+\cos \left(\frac{\mathrm{\pi }}{7}\right)+\mathrm{I}\sin \left(\frac{\mathrm{\pi }}{7}\right)\right)\left(x+\cos \left(\frac{3\mathrm{\pi }}{7}\right)+\mathrm{I}\sin \left(\frac{3\mathrm{\pi }}{7}\right)\right)\left(x-\cos \left(\frac{2\mathrm{\pi }}{7}\right)+\mathrm{I}\sin \left(\frac{2\mathrm{\pi }}{7}\right)\right)$

$\mathrm{IsCyclotomicPolynomial}\left(q\,x\right)$

$\mathrm{true}$

The NumberTheory[IsCyclotomicPolynomial] command was introduced in Maple 2016.

For more information on Maple 2016 changes, see Updates in Maple 2016.