Complex Algebra/Examples/z^6 + z^3 + 1

Example of Complex Algebra

 * $z^6 + z^3 + 1 = \paren {z^2 - 2 z \cos \dfrac {2 \pi} 9 + 1} \paren {z^2 - 2 z \cos \dfrac {4 \pi} 9 + 1} \paren {z^2 - 2 z \cos \dfrac {8 \pi} 9 + 1}$

Proof
From Sum of Geometric Progression or Difference of Two Cubes:
 * $z^6 + z^3 + 1 = \dfrac {z^9 - 1} {z^3 - 1}$

Then from Factorisation of $x^{2 n + 1} - 1$ in Real Domain:
 * $z^9 - 1 = \paren {z - 1} \displaystyle \prod_{k \mathop = 1}^4 \paren {z^2 - 2 \cos \dfrac {2 \pi k} 9 + 1}$

and:

Thus:
 * $\paren {z^3 - 1} \paren {z^6 + z^3 + 1} = \paren {\paren {z - 1} \paren {z^2 - 2 \cos \dfrac {6 \pi} 9 + 1} } \paren {\paren {z^2 - 2 \cos \dfrac {2 \pi} 9 + 1} \paren {z^2 - 2 \cos \dfrac {4 \pi} 9 + 1} \paren {z^2 - 2 \cos \dfrac {8 \pi} 9 + 1} }$

from which:
 * $\paren {z^6 + z^3 + 1} = \paren {z^2 - 2 \cos \dfrac {2 \pi} 9 + 1} \paren {z^2 - 2 \cos \dfrac {4 \pi} 9 + 1} \paren {z^2 - 2 \cos \dfrac {8 \pi} 9 + 1}$