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  "body": "\n## Statement\n\nFor every integer $n \\ge 1$, the [[cyclotomic polynomial|0W0KVBM54S]] $\\Phi_n$ is [[irreducible|00X2A9G9W3]] in $\\mathbb{Q}[x]$.\n\n## Remarks\n\nFor $n = p$ prime, [[Eisenstein's criterion|02BPKD7TMD]] applied to $\\Phi_p(x + 1)$ with the prime $p$ proves the statement.\nThe general case is harder. One proof shows that if $\\zeta$ is a root of an irreducible factor of $\\Phi_n$, then so is $\\zeta^p$ for each prime $p \\nmid n$.\nA consequence: $\\Phi_n$ is the [[minimal polynomial|0GK1WB9R98]] over $\\mathbb{Q}$ of each primitive $n$-th root of unity. So the degree of $\\mathbb{Q}(e^{2\\pi i/n})$ over $\\mathbb{Q}$ is $\\varphi(n)$.\n\n## Explanations\n\n### L1 human\n\nThe polynomial $\\Phi_n$ cannot be split into smaller polynomials with rational coefficients.\nSo all the primitive $n$-th roots of unity are conjugate over the rationals: no rational polynomial equation separates them.\n",
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