Complex Roots of Unity/Examples/Cube Roots/Conjugate Form

Example of Complex Roots of Unity

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 $\omega$.

We have that the Cube Roots of Unity can be expressed as:

$U_3 = \set {1, \omega, \omega^2}$

Then:

$\ds \omega^2$

$=$

$\ds \frac {\omega^3} \omega$

$\ds $

$=$

$\ds \frac 1 \omega$

$\ds $

$=$

$\ds \frac {\overline \omega} {\omega \overline \omega}$

$\ds $

$=$

$\ds \frac {\overline \omega} {\cmod {\omega}^2}$

$\ds $

$=$

$\ds \overline \omega$

$\blacksquare$