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  "body": "\n## Definition\n\nLet $n$ be a positive integer.\nA 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 $1 \\le k \u003c n$.\n\n## Remarks\n\nEquivalently, $\\zeta$ has order exactly $n$ in the multiplicative group $\\mathbb{C} \\setminus \\{0\\}$.\nIf $\\zeta$ 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|0S61N75H45]].\nThe 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.\n\n## Explanations\n\n### L1 human\n\nThe number $i$ is a primitive $4$th root of unity: $i, i^2 = -1, i^3 = -i$ are not $1$, and $i^4 = 1$.\nThe number $-1$ is a $4$th root of unity, but not a primitive one, because already $(-1)^2 = 1$.\n",
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