Primitive root of unity

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.