Editing Factorization (section)

====Roots of unity====
The {{mvar|n}}th [[roots of unity]] are the [[complex number]]s each of which is a [[zero of a function|root]] of the polynomial <math>x^n-1.</math> They are thus the numbers 
<math display="block">e^{2ik\pi/n} = \cos \tfrac{2\pi k}n + i\sin \tfrac{2\pi k} n </math>
for <math>k=0, \ldots, n-1.</math>

It follows that for any two expressions {{mvar|E}} and {{mvar|F}}, one has:
<math display="block">E^n-F^n = (E-F) \prod_{k=1}^{n-1} \left(E-F e^{2ik\pi/n}\right)</math>
<math display="block">E^n + F^n = \prod_{k=0}^{n-1} \left(E-F e^{(2k+1)i\pi/n}\right) \qquad \text{if } n \text{ is even}</math>
<math display="block">E^{n}+F^{n}=(E+F) \prod_{k=1}^{n-1}\left(E+F e^{2ik\pi/n}\right) \qquad \text{if } n \text{ is odd}</math>

If {{mvar|E}} and {{mvar|F}} are real expressions, and one wants real factors, one has to replace every pair of [[complex conjugate]] factors by its product. As the complex conjugate of <math>e^{i\alpha}</math> is <math>e^{-i\alpha},</math> and 
<math display="block">\left(a-be^{i\alpha}\right) \left(a-be^{-i\alpha}\right)=
a^2 - ab\left(e^{i\alpha}+e^{-i\alpha}\right) + b^2e^{i\alpha}e^{-i\alpha} =
a^2 - 2ab\cos\,\alpha + b^2,  </math>
one has the following real factorizations (one passes from one to the other by changing {{mvar|k}} into {{math|''n'' − ''k''}} or {{math|''n'' + 1 − ''k''}}, and applying the usual [[trigonometric formulas]]:
<math display="block">\begin{align}
E^{2n}-F^{2n}&= 
(E-F)(E+F)\prod_{k=1}^{n-1} \left(E^2-2EF \cos\,\tfrac{k\pi}n +F^2\right)\\
&=(E-F)(E+F)\prod_{k=1}^{n-1} \left(E^2+2EF \cos\,\tfrac{k\pi}n +F^2\right)
\end{align}</math>
<math display="block"> \begin{align}
E^{2n} + F^{2n} &= 
\prod_{k=1}^n \left(E^2 + 2EF\cos\,\tfrac{(2k-1)\pi}{2n}+F^2\right)\\
&=\prod_{k=1}^n \left(E^2 - 2EF\cos\,\tfrac{(2k-1)\pi}{2n}+F^2\right)
\end{align}</math>

The [[cosine]]s that appear in these factorizations are [[algebraic number]]s, and may be expressed in terms of [[nth root|radicals]] (this is possible because their [[Galois group]] is cyclic); however, these radical expressions are too complicated to be used, except for low values of {{mvar|n}}. For example,
<math display="block"> a^4 + b^4 = (a^2 - \sqrt 2 ab + b^2)(a^2 + \sqrt 2 ab + b^2).</math>
<math display="block"> a^5 - b^5 = (a - b) \left(a^2 + \frac{1-\sqrt 5}2 ab + b^2\right) \left(a^2 +\frac{1+\sqrt 5}2 ab + b^2\right),</math>
<math display="block"> a^5 + b^5 = (a + b) \left(a^2 - \frac{1-\sqrt 5}2 ab + b^2\right) \left(a^2 -\frac{1+\sqrt 5}2 ab + b^2\right),</math>

Often one wants a factorization with rational coefficients. Such a factorization involves [[cyclotomic polynomial]]s. To express rational factorizations of sums and differences or powers, we need a notation for the [[homogenization of a polynomial]]: if <math>P(x)=a_0x^n+a_ix^{n-1} +\cdots +a_n,</math> its ''homogenization'' is the [[bivariate polynomial]] <math>\overline P(x,y)=a_0x^n+a_ix^{n-1}y +\cdots +a_ny^n.</math> Then, one has
<math display="block">E^n-F^n=\prod_{k\mid n}\overline Q_n(E,F),</math>
<math display="block">E^n+F^n=\prod_{k\mid 2n,k\not\mid n}\overline Q_n(E,F),</math>
where the products are taken over all divisors of {{mvar|n}}, or all divisors of {{math|2''n''}} that do not divide {{mvar|n}}, and <math>Q_n(x)</math> is the {{mvar|n}}th cyclotomic polynomial.

For example, 
<math display="block">a^6-b^6= \overline Q_1(a,b)\overline Q_2(a,b)\overline Q_3(a,b)\overline Q_6(a,b)=(a-b)(a+b)(a^2-ab+b^2)(a^2+ab+b^2),</math>
<math display="block">a^6+b^6=\overline Q_4(a,b)\overline Q_{12}(a,b) = (a^2+b^2)(a^4-a^2b^2+b^4),</math>
since the divisors of 6 are 1, 2, 3, 6, and the divisors of 12 that do not divide 6 are 4 and 12.