Editing Cyclotomic polynomial (section)

===Easy cases for computation===
As noted above, if {{math|1=''n'' = ''p''}} is a prime number, then

:<math>\Phi_p(x) = 1+x+x^2+\cdots+x^{p-1}=\sum_{k=0}^{p-1}x^k\;.</math>

If ''n'' is an odd integer greater than one, then 

:<math>\Phi_{2n}(x) = \Phi_n(-x)\;.</math>

In particular, if {{math|1=''n'' = 2''p''}} is twice an odd prime, then (as noted above)

:<math>\Phi_{2p}(x) = 1-x+x^2-\cdots+x^{p-1}=\sum_{k=0}^{p-1}(-x)^k\;.</math>

If {{math|1=''n'' = ''p<sup>m</sup>''}} is a [[prime power]] (where ''p'' is prime), then

:<math>\Phi_{p^m}(x) = \Phi_p(x^{p^{m-1}}) =\sum_{k=0}^{p-1}x^{kp^{m-1}}\;.</math>

More generally, if {{math|1=''n'' = ''p<sup>m</sup>r''}} with {{math|''r''}} [[relatively prime]] to {{math|''p''}}, then 

:<math>\Phi_{p^mr}(x) = \Phi_{pr}(x^{p^{m-1}})\;.</math>

These formulas may be applied repeatedly to get a simple expression for any cyclotomic polynomial <math>\Phi_n(x)</math> in terms of a cyclotomic polynomial of [[square-free number|square free]] index: If {{math|''q''}} is the [[Product (mathematics)|product]] of the prime divisors of {{math|''n''}} (its [[Radical of an integer|radical]]), then<ref>{{citation
 | last = Cox | first = David A. | author-link = David A. Cox  | contribution = Exercise 12  | doi = 10.1002/9781118218457 | edition = 2nd  | isbn = 978-1-118-07205-9 | page = 237 | publisher = John Wiley & Sons  | title = Galois Theory | year = 2012}}.</ref>

:<math>\Phi_n(x) = \Phi_q(x^{n/q})\;.</math>

This allows formulas to be given for the {{math|''n''}}th cyclotomic polynomial when {{math|''n''}} has at most one odd prime factor: If {{math|''p''}} is an odd prime number, and {{tmath|\ell}} and {{math|''m''}} are positive integers, then
:<math>\Phi_{2^m}(x) = x^{2^{m-1}}+1\;,</math>
:<math>\Phi_{p^m}(x) = \sum_{j=0}^{p-1}x^{jp^{m-1}}\;,</math>
:<math>\Phi_{2^\ell p^m}(x) = \sum_{j=0}^{p-1}(-1)^jx^{j2^{\ell-1}p^{m-1}}\;.</math>

For other values of {{math|''n''}}, the computation of the {{math|''n''}}th cyclotomic polynomial is similarly reduced to that of  <math>\Phi_q(x),</math> where {{math|''q''}} is the product of the distinct odd prime divisors of {{math|''n''}}. To deal with this case, one has that, for {{math|''p''}} prime and not dividing {{math|''n''}},<ref name="WolframCyclotomic">{{MathWorld |title=Cyclotomic Polynomial |id=CyclotomicPolynomial|mode=cs2}}</ref>

:<math>\Phi_{np}(x)=\Phi_{n}(x^p)/\Phi_n(x)\;.</math>