Roots of Complex Number/Examples/z^5 + 1 = 0

Theorem
The roots of the polynomial:
 * $z^5 + 1 = 0$

are:
 * $\set {\cos \dfrac \pi 5 \pm i \sin \dfrac \pi 5, \cos \dfrac {3 \pi} 5 \pm i \sin \dfrac {3 \pi} 5, -1}$

Proof
From Factorisation of $z^n + 1$:


 * $z^5 + 1 = \ds \prod_{k \mathop = 0}^4 \paren {z - \exp \dfrac {\paren {2 k + 1} i \pi} 5}$

Thus:
 * $z = \set {\exp \dfrac {\paren {2 k + 1} i \pi} 5}$