Complex Algebra/Examples/z^8 + 1

Example of Complex Algebra

 * $z^8 + 1 = \paren {z^2 - 2 z \cos \dfrac \pi 8 + 1} \paren {z^2 - 2 z \cos \dfrac {3 \pi} 8 + 1} \paren {z^2 - 2 z \cos \dfrac {5 \pi} 8 + 1} \paren {z^2 - 2 z \cos \dfrac {7 \pi} 8 + 1}$

Proof
From Roots of $z^8 + 1 = 0$ and the corollary to the Polynomial Factor Theorem:
 * $z^8 + 1 = \ds \prod_{k \mathop = 0}^7 \paren {z - \paren {\cos \dfrac {\paren {2 k + 1} \pi} 8 + i \sin \dfrac {\paren {2 k + 1} \pi} 8} }$

Hence:

Hence the result.