Complex Roots of Unity/Examples/Cube Roots

Example of Complex Roots of Unity

The are the elements of the set:

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

They are:

$\ds $

$\, \ds e^{0 i \pi / 3} \, $

$\, \ds = \, $

$\ds 1$

$\ds \omega$

$=$

$\, \ds e^{2 i \pi / 3} \, $

$\, \ds = \, $

$\ds -\frac 1 2 + \frac {i \sqrt 3} 2$

$\ds \omega^2$

$=$

$\, \ds e^{4 i \pi / 3} \, $

$\, \ds = \, $

$\ds -\frac 1 2 - \frac {i \sqrt 3} 2$

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

$\ds z^3 - 1$

$=$

$\ds \paren {z - 1} \paren {z^2 + z + 1}$

Difference of Two Cubes/Corollary

$\ds \leadsto \ \ $

$\ds z$

$=$

$\ds 1$

$\, \ds \text { or } \, $

$\ds z^2 + z + 1$

$=$

$\ds 0$

Then:

$\ds z^2 + z + 1$

$=$

$\ds 0$

$\ds \leadsto \ \ $

$\ds z$

$=$

$\ds \dfrac {-1 \pm \sqrt {1^2 - 4 \times 1 \times 1} } {2 \times 1}$

Quadratic Formula

$\ds $

$=$

$\ds -\frac 1 2 \pm i \frac {\sqrt 3} 2$

simplifying

$\blacksquare$

Sources

1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 3$. Roots of Unity
1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Introduction
1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): cube root of unity
1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): root of unity
2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): cube root of unity
2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): root of unity
2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): cube root of unity