Roots of Complex Number/Corollary/Examples/Cube Roots
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Example of Roots of Complex Number: Corollary
Let $z \in \C$ be a complex number.
Let $z \ne 0$.
Let $w$ be one of the (complex) cube roots of $z$.
Then the complete set of (complex) cube roots of $z$ is:
- $\set {w, w \omega, w \omega^2}$
where:
- $\omega = e^{2 \pi / 3} = -\dfrac 1 2 + \dfrac {i \sqrt 3} 2$
Proof
From Primitive Complex Cube Roots of Unity, $\omega$ is a primitive root of $1$.
Hence the result from Roots of Complex Number: Corollary.
$\blacksquare$
Sources
- 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Introduction