Complex Algebra/Examples/z^5 + 1

Example of Complex Algebra

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

Proof

From Factorisation of $z^{2 n + 1} + 1$ in Real Domain:

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

Hence the result.

$\blacksquare$

Sources

• 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 3$. Roots of Unity: Exercise $9$