Cyclotomic Polynomial - Maple Help

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NumberTheory

 CyclotomicPolynomial
 minimal polynomials of primitive roots of unity with rational coefficients

Calling Sequence

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

Parameters

 n - positive integer x - name

Description

 • 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 $\mathrm{\Phi }\left(8,x\right)$.

Examples

 > $\mathrm{with}\left(\mathrm{NumberTheory}\right):$
 > $\mathrm{CyclotomicPolynomial}\left(1,x\right)$
 ${x}{-}{1}$ (1)
 > $\mathrm{\Phi }\left(2,x\right)$
 ${x}{+}{1}$ (2)

The one hundred and fifth cyclotomic polynomial is the first with a coefficient greater than 1.

 > $\mathrm{CyclotomicPolynomial}\left(105,x\right)$
 ${{x}}^{{48}}{+}{{x}}^{{47}}{+}{{x}}^{{46}}{-}{{x}}^{{43}}{-}{{x}}^{{42}}{-}{2}{}{{x}}^{{41}}{-}{{x}}^{{40}}{-}{{x}}^{{39}}{+}{{x}}^{{36}}{+}{{x}}^{{35}}{+}{{x}}^{{34}}{+}{{x}}^{{33}}{+}{{x}}^{{32}}{+}{{x}}^{{31}}{-}{{x}}^{{28}}{-}{{x}}^{{26}}{-}{{x}}^{{24}}{-}{{x}}^{{22}}{-}{{x}}^{{20}}{+}{{x}}^{{17}}{+}{{x}}^{{16}}{+}{{x}}^{{15}}{+}{{x}}^{{14}}{+}{{x}}^{{13}}{+}{{x}}^{{12}}{-}{{x}}^{{9}}{-}{{x}}^{{8}}{-}{2}{}{{x}}^{{7}}{-}{{x}}^{{6}}{-}{{x}}^{{5}}{+}{{x}}^{{2}}{+}{x}{+}{1}$ (3)
 > $\mathrm{Totient}\left(105\right)$
 ${48}$ (4)
 > $p≔\mathrm{CyclotomicPolynomial}\left(7,x\right)$
 ${p}{≔}{{x}}^{{6}}{+}{{x}}^{{5}}{+}{{x}}^{{4}}{+}{{x}}^{{3}}{+}{{x}}^{{2}}{+}{x}{+}{1}$ (5)
 > $r≔\left[\mathrm{solve}\left(p=0,x\right)\right]$
 ${r}{≔}\left[{\mathrm{cos}}{}\left(\frac{{2}{}{\mathrm{\pi }}}{{7}}\right){+}{I}{}{\mathrm{sin}}{}\left(\frac{{2}{}{\mathrm{\pi }}}{{7}}\right){,}{-}{\mathrm{cos}}{}\left(\frac{{3}{}{\mathrm{\pi }}}{{7}}\right){+}{I}{}{\mathrm{sin}}{}\left(\frac{{3}{}{\mathrm{\pi }}}{{7}}\right){,}{-}{\mathrm{cos}}{}\left(\frac{{\mathrm{\pi }}}{{7}}\right){+}{I}{}{\mathrm{sin}}{}\left(\frac{{\mathrm{\pi }}}{{7}}\right){,}{-}{\mathrm{cos}}{}\left(\frac{{\mathrm{\pi }}}{{7}}\right){-}{I}{}{\mathrm{sin}}{}\left(\frac{{\mathrm{\pi }}}{{7}}\right){,}{-}{\mathrm{cos}}{}\left(\frac{{3}{}{\mathrm{\pi }}}{{7}}\right){-}{I}{}{\mathrm{sin}}{}\left(\frac{{3}{}{\mathrm{\pi }}}{{7}}\right){,}{\mathrm{cos}}{}\left(\frac{{2}{}{\mathrm{\pi }}}{{7}}\right){-}{I}{}{\mathrm{sin}}{}\left(\frac{{2}{}{\mathrm{\pi }}}{{7}}\right)\right]$ (6)
 > $\mathrm{plots}:-\mathrm{complexplot}\left(r,\mathrm{style}=\mathrm{point}\right)$

Compatibility

 • The NumberTheory[CyclotomicPolynomial] command was introduced in Maple 2016.
 • For more information on Maple 2016 changes, see Updates in Maple 2016.