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  "body": "\n## Definition\n\nLet $K$ be a field and let $n \\ge 1$ be an integer that is not divisible by the characteristic of $K$.\nA [[field extension|0WQ1A8SHTZ]] $L/K$ is an *$n$-th cyclotomic extension* if both conditions hold:\n\n1. $L$ contains a [[primitive $n$-th root of unity|0J817RTVG6]] $\\zeta_n$;\n2. $L = K(\\zeta_n)$.\n\nEquivalently, $L$ is a [[splitting field|02NZZ1J0FH]] of $x^n - 1$ over $K$.\nMore generally, let $S$ be a set of positive integers.\nThen $L/K$ is an *$S$-cyclotomic extension* if $L$ contains a primitive $n$-th root of unity for each $n \\in S$.\nMoreover $L$ must be generated over $K$ by the [[roots of unity|0S61N75H45]] of these orders.\n\n## Remarks\n\nExample: $\\Q(i)/\\Q$ is a $4$-th cyclotomic extension, since $i$ has order $4$ in $\\C^\\times$ and $\\Q(i)$ is generated by $i$.\nNon-example: $\\Q(\\sqrt[3]{2})/\\Q$ is not a $3$-rd cyclotomic extension, since $\\Q(\\sqrt[3]{2})$ is a subfield of $\\R$ and contains no primitive cube root of unity.\nThe two forms agree because the roots of $x^n - 1$ are the powers of $\\zeta_n$.\nAn $n$-th cyclotomic extension is Galois, and $\\sigma \\mapsto k$ with $\\sigma(\\zeta_n) = \\zeta_n^k$ is an injective homomorphism $\\Gal(L/K) \\to (\\Z/n\\Z)^\\times$.\nFor $K = \\Q$ this map is an isomorphism, and $L$ is the cyclotomic field $\\Q(\\zeta_n)$.\n",
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