The CyclotomicPolynomial('n', 'x') command computes the nth cyclotomic polynomial in x.

The roots of the nth cyclotomic polynomial are exactly the nth primitive roots of unity.

The degree of the nth cyclotomic polynomial is given by Euler's totient function, NumberTheory[Totient].

Phi is an alias for CyclotomicPolynomial.

You can enter the command Phi using either the 1-D or 2-D calling sequence. For example, Phi(8, x) is equivalent to $\mathrm{\Phi }\left(8\,x\right)$.

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

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

$x-1$

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

$x+1$

$\mathrm{CyclotomicPolynomial}\left(105\,x\right)$

$x^{48}+x^{47}+x^{46}-x^{43}-x^{42}-2x^{41}-x^{40}-x^{39}+x^{36}+x^{35}+x^{34}+x^{33}+x^{32}+x^{31}-x^{28}-x^{26}-x^{24}-x^{22}-x^{20}+x^{17}+x^{16}+x^{15}+x^{14}+x^{13}+x^{12}-x^9-x^8-2x^7-x^6-x^5+x^2+x+1$

$\mathrm{Totient}\left(105\right)$

$48$

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

$p≔x^6+x^5+x^4+x^3+x^2+x+1$

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

$r≔\left[\cos \left(\frac{2\mathrm{\pi }}{7}\right)+\mathrm{I}\sin \left(\frac{2\mathrm{\pi }}{7}\right)\,-\cos \left(\frac{3\mathrm{\pi }}{7}\right)+\mathrm{I}\sin \left(\frac{3\mathrm{\pi }}{7}\right)\,-\cos \left(\frac{\mathrm{\pi }}{7}\right)+\mathrm{I}\sin \left(\frac{\mathrm{\pi }}{7}\right)\,-\cos \left(\frac{\mathrm{\pi }}{7}\right)-\mathrm{I}\sin \left(\frac{\mathrm{\pi }}{7}\right)\,-\cos \left(\frac{3\mathrm{\pi }}{7}\right)-\mathrm{I}\sin \left(\frac{3\mathrm{\pi }}{7}\right)\,\cos \left(\frac{2\mathrm{\pi }}{7}\right)-\mathrm{I}\sin \left(\frac{2\mathrm{\pi }}{7}\right)\right]$

$\mathrm{plots}:-\mathrm{complexplot}\left(r\,\mathrm{style}=\mathrm{point}\right)$

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

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