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  "body": "\n## Summary\n\nThe $n$-th cyclotomic field is the field $\\Q(\\zeta_n)$ obtained by adjoining to $\\Q$ a [[primitive $n$-th root of unity|0EHM5ZP5QH]] $\\zeta_n$.\nIt is a [[Galois extension|0QMZAQ6QQ4]] of $\\Q$ of degree $\\varphi(n)$ whose Galois group is the [[group of units of $\\Z/n\\Z$|0F8X8ZRQ53]].\nBy the [[Kronecker–Weber theorem|AW00HPFB54]], every finite abelian extension of $\\Q$ lies in some cyclotomic field.\nCyclotomic fields were studied by Kummer in his work on Fermat's Last Theorem, which led to the notion of a [[regular prime|B4VK33PCXZ]].\n\n## Explanations\n\n### L1 human\n\nThe $n$-th roots of unity are the $n$ corners of a regular $n$-gon inscribed in the unit circle, with one corner at $1$.\nThe cyclotomic field is what you get by adjoining these corners to the rational numbers.\nFor $n = 4$ the corners are $1, i, -1, -i$, and the field is $\\Q(i)$.\nFor $n = 3$ the field is $\\Q(\\sqrt{-3})$, since $e^{2\\pi i/3} = (-1 + \\sqrt{-3})/2$.\nThe symmetries of the field are the maps $\\zeta_n \\mapsto \\zeta_n^a$ with $a$ coprime to $n$.\n",
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