exp(2 pi i/n) is a primitive root of unity

Statement

Let $n$ be a positive integer. Then $e^{2 \pi i / n}$ is a primitive root of unity.

Remarks

Proof: for $1 \le k < n$, the number $\zeta^k = e^{2 \pi i k / n}$ has argument $2 \pi k / n$, which is strictly between $0$ and $2\pi$. Thus $\zeta^k \neq 1$. For $n = 4$ this number is $i$, and for $n = 2$ it is $-1$.

Explanations

L1 human

The first root of unity after $1$, going counterclockwise around the circle, is a primitive one. Its powers walk around the circle step by step and visit every $n$th root of unity before they return to $1$.