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  "body": "\n## Definition\n\nLet $n \\ge 1$ be an integer.\nThe $n$-th cyclotomic polynomial is\n$$\\Phi_n(x) = \\prod_{\\substack{1 \\le k \\le n \\\\ \\gcd(k, n) = 1}} \\left(x - e^{2 \\pi i k / n}\\right) \\in \\mathbb{C}[x].$$\nIts roots are the primitive $n$-th roots of unity: the complex numbers $\\zeta$ with $\\zeta^n = 1$ and $\\zeta^m \\neq 1$ for $0 \u003c m \u003c n$.\n\n## Remarks\n\nThe polynomial $\\Phi_n$ is [[monic|0BDX4MBKT4]] of degree $\\varphi(n)$, where $\\varphi$ is Euler's totient function.\nFor each $n \\ge 1$, $x^n - 1 = \\prod_{d \\mid n} \\Phi_d(x)$. By induction on $n$, this shows that $\\Phi_n$ has integer coefficients.\nThe first values are $\\Phi_1 = x - 1$, $\\Phi_2 = x + 1$, $\\Phi_3 = x^2 + x + 1$, $\\Phi_4 = x^2 + 1$ and $\\Phi_6 = x^2 - x + 1$.\nFor a prime $p$, $\\Phi_p = x^{p-1} + x^{p-2} + \\dots + x + 1$.\n\n## Explanations\n\n### L1 human\n\nThe $n$-th roots of unity are the corners of a regular $n$-gon on the unit circle.\nThe cyclotomic polynomial $\\Phi_n$ collects only the corners that are not roots of unity of a smaller order.\nThe name means \"circle dividing\".\n",
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