Powers of Primitive Complex Root of Unity form Complete Set

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