Group of nth roots of unity

Description

Let $n$ be a positive integer. The group of $n$th roots of unity is the set $\mu_n = \{z \in \mathbb{C} : z^n = 1\}$ with complex multiplication. It is a finite abelian group of order $n$, and its elements are $e^{2\pi i k / n}$ for $k = 0, 1, \ldots, n - 1$. It is cyclic, and the element $e^{2\pi i / n}$ generates it.

Remarks

The group $\mu_n$ is a subgroup of the circle group. For $n = 4$, the group is $\{1, i, -1, -i\}$. A generator of $\mu_n$ is called a primitive $n$th root of unity, and there are $\varphi(n)$ of them, where $\varphi$ is Euler's totient function.

Explanations

L1 human

The $n$th roots of unity are $n$ points on the unit circle that cut it into $n$ equal arcs. Multiplying two of them adds their angles, so the product is again one of these points. They behave like the rotations of a regular polygon with $n$ sides.