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  "body": "\n## Description\n\nLet $n$ be a positive integer.\nThe group of $n$th roots of unity is the set $\\mu_n = \\{z \\in \\mathbb{C} : z^n = 1\\}$ with complex multiplication.\nIt is a finite [[abelian group|082GHNPJ53]] of order $n$, and its elements are $e^{2\\pi i k / n}$ for $k = 0, 1, \\ldots, n - 1$.\nIt is cyclic, and the element $e^{2\\pi i / n}$ generates it.\n\n## Remarks\n\nThe group $\\mu_n$ is a subgroup of the [[circle group|04EAFVFTF7]].\nFor $n = 4$, the group is $\\{1, i, -1, -i\\}$.\nA 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.\n\n## Explanations\n\n### L1 human\n\nThe $n$th roots of unity are $n$ points on the unit circle that cut it into $n$ equal arcs.\nMultiplying two of them adds their angles, so the product is again one of these points.\nThey behave like the rotations of a regular polygon with $n$ sides.\n",
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