Primitive root of unity

Definition

Let $n$ be a positive integer. A complex number $\zeta$ is a primitive $n$th root of unity when $\zeta^n = 1$ and $\zeta^k \neq 1$ for each integer $k$ with $1 \le k < n$.

Remarks

Equivalently, $\zeta$ has order exactly $n$ in the multiplicative group $\mathbb{C} \setminus \{0\}$. If $\zeta$ is a primitive $n$th root of unity, then $1, \zeta, \zeta^2, \dots, \zeta^{n-1}$ are all the $n$th roots of unity. The number $e^{2 \pi i k / n}$ is primitive exactly when $\gcd(k, n) = 1$. Thus there are $\varphi(n)$ primitive $n$th roots of unity, where $\varphi$ is Euler's totient function.

Explanations

L1 human

The number $i$ is a primitive $4$th root of unity: $i, i^2 = -1, i^3 = -i$ are not $1$, and $i^4 = 1$. The number $-1$ is a $4$th root of unity, but not a primitive one, because already $(-1)^2 = 1$.