First Complex Root of Unity is Primitive

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 $\alpha_1 = \exp \paren {\dfrac {2 \pi i} n}$ denote the first complex root of unity.

Then $\alpha_1$ is a primitive complex root of unity.

Proof

From Condition for Complex Root of Unity to be Primitive:

$\gcd \set {n, k} = 1$

where:

$\alpha_k = \exp \paren {\dfrac {2 k \pi i} n}$

Here we have:

$k = 1$

and:

$\gcd \set {n, 1} = 1$

Hence the result.

$\blacksquare$

Sources

• 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 3$. Roots of Unity