Powers of Primitive Complex Root of Unity form Complete Set
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Theorem
Let $n \in \Z_{>0}$ be a (strictly) positive integer.
Let $U_n$ denote the complex $n$th roots of unity:
- $U_n = \set {z \in \C: z^n = 1}$
Let $\alpha_k = \exp \paren {\dfrac {2 k \pi i} n}$ denote the $k$th complex root of unity.
Let $\alpha_k$ be a primitive complex root of unity.
Let $V_k = \set { {\alpha_k}^r: r \in \set {0, 1, \ldots, n - 1} }$.
Then:
- $V_k = U_n$
That is, $V_k = \set { {\alpha_k}^r: r \in \set {0, 1, \ldots, n - 1} }$ forms the complete set of complex $n$th roots of unity.
Proof
From Roots of Unity under Multiplication form Cyclic Group, $\struct {U_n, \times}$ is a group.
The result follows from Power of Generator of Cyclic Group is Generator iff Power is Coprime with Order.
$\blacksquare$
Sources
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 3$. Roots of Unity