Roots of Complex Number/Corollary

Theorem
Let $z := \polar {r, \theta}$ be a complex number expressed in polar form, such that $z \ne 0$.

Let $n \in \Z_{>0}$ be a (strictly) positive integer.

Let $w$ be one of the complex $n$th roots of $z$.

Then the $n$th roots of $z$ are given by:
 * $z^{1 / n} = \set {w \epsilon^k: k \in \set {1, 2, \ldots, n - 1} }$

where $\epsilon$ is a primitive $n$th root of unity.

Proof
By definition of primitive complex $n$th root of unity:
 * $\omega = e^{2 m i \pi k}$

for some $m \in \Z: 1 \le m < n$.

Thus:

This demonstrates that $w \omega^k$ is one of the complex $n$th roots of $z$.

All of the complex $n$th roots of unity are represented by powers of $\omega$.

Thus it follows from Roots of Complex Number that:
 * $z^{1 / n} = \set {w \omega^k: k \in \set {1, 2, \ldots, n - 1} }$

are the $n$ complex $n$th roots of $z$.