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.