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  "body": "\n## Definition\n\nLet $n \\ge 1$ be an integer and let $\\zeta_n = e^{2\\pi i/n} \\in \\C$, a [[primitive $n$-th root of unity|0EHM5ZP5QH]].\nThe *$n$-th cyclotomic field* is the subfield $\\Q(\\zeta_n)$ of $\\C$ generated by $\\zeta_n$ over $\\Q$.\nA *cyclotomic field* is a field of this form.\n\n## Remarks\n\nThe minimal polynomial of $\\zeta_n$ over $\\Q$ is the [[cyclotomic polynomial|08KEYFAH66]] $\\Phi_n$, which is irreducible over $\\Q$ by the [[irreducibility of cyclotomic polynomials|0SK1VHZ5R2]].\nSo $[\\Q(\\zeta_n) : \\Q] = \\varphi(n)$, where $\\varphi$ is [[Euler's totient function|0DAMHCT17R]], and $\\Q(\\zeta_n)$ is an [[algebraic number field|MWBDFS2KJ5]].\nExample: for $n = 4$, $\\zeta_4 = i$ and $\\Phi_4 = x^2 + 1$, so $\\Q(\\zeta_4) = \\Q(i)$ has degree $\\varphi(4) = 2$.\nNon-example: $\\Q(\\sqrt[3]{2})$ is not a cyclotomic field: every cyclotomic field is a Galois extension of $\\Q$, but $\\Q(\\sqrt[3]{2}) \\subset \\R$ does not contain the non-real roots of $x^3 - 2$.\nThe Galois group of $\\Q(\\zeta_n)/\\Q$ is isomorphic to $(\\Z/n\\Z)^{\\times}$, with $a$ acting by $\\zeta_n \\mapsto \\zeta_n^a$.\nThe ring of integers of $\\Q(\\zeta_n)$ is $\\Z[\\zeta_n]$.\nFor odd $n$, $\\Q(\\zeta_{2n}) = \\Q(\\zeta_n)$, because $-\\zeta_n$ is a primitive $2n$-th root of unity.\n",
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