Complex Roots of Unity/Examples/Cube Roots

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Example of Complex Roots of Unity

The complex cube roots of unity are the elements of the set:

$U_3 = \set {z \in \C: z^3 = 1}$


They are:

\(\displaystyle \) \(\) \(\, \displaystyle e^{0 i \pi / 3} \, \) \(\, \displaystyle =\, \) \(\displaystyle 1\)
\(\displaystyle \omega\) \(=\) \(\, \displaystyle e^{2 i \pi / 3} \, \) \(\, \displaystyle =\, \) \(\displaystyle -\frac 1 2 + \frac {i \sqrt 3} 2\)
\(\displaystyle \omega^2\) \(=\) \(\, \displaystyle e^{4 i \pi / 3} \, \) \(\, \displaystyle =\, \) \(\displaystyle -\frac 1 2 - \frac {i \sqrt 3} 2\)


The notation $\omega$ for, specifically, the complex cube roots of unity is conventional.


Conjugate Form

The Cube Roots of Unity can be expressed in the form:

$U_3 = \set {1, \omega, \overline \omega}$

where:

$\omega = -\dfrac 1 2 + \dfrac {i \sqrt 3} 2$
$\overline \omega$ denotes the complex conjugate of $\omega$.


Proof

\(\displaystyle z^3 - 1\) \(=\) \(\displaystyle \paren {z - 1} \paren {z^2 + z + 1}\) Difference of Two Cubes/Corollary
\(\displaystyle \leadsto \ \ \) \(\displaystyle z\) \(=\) \(\displaystyle 1\)
\(\, \displaystyle \text { or } \, \) \(\displaystyle z^2 + z + 1\) \(=\) \(\displaystyle 0\)

Then:

\(\displaystyle z^2 + z + 1\) \(=\) \(\displaystyle 0\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle z\) \(=\) \(\displaystyle \dfrac {-1 \pm \sqrt {1^2 - 4 \times 1 \times 1} } {2 \times 1}\) Quadratic Formula
\(\displaystyle \) \(=\) \(\displaystyle -\frac 1 2 \pm i \frac {\sqrt 3} 2\) simplifying

$\blacksquare$


Sources