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  "body": "\n## Summary\n\nA cyclotomic extension of a field $K$ is obtained by adjoining [[roots of unity|0S61N75H45]] to $K$.\nThe extension $K(\\zeta_n)/K$ generated by a [[primitive $n$-th root of unity|0J817RTVG6]] is Galois, and its Galois group embeds in $(\\Z/n\\Z)^\\times$, so it is [[abelian|EQDEXBWGJH]].\nOver $\\Q$ the embedding is an isomorphism, because the [[cyclotomic polynomials|0W0KVBM54S]] are [[irreducible over the rationals|0SK1VHZ5R2]].\nThe fields $\\Q(\\zeta_n)$ are the cyclotomic fields of number theory.\n\n## Explanations\n\n### L1 human\n\nThe $n$-th roots of unity are the $n$ complex numbers whose $n$-th power is $1$; they sit at the corners of a regular $n$-gon on the unit circle.\nAdjoining one primitive root $\\zeta_n$ gives all the others, because they are powers of it.\nEach symmetry of $\\Q(\\zeta_n)$ sends $\\zeta_n$ to a power $\\zeta_n^k$ with $k$ coprime to $n$.\nFor $n = 4$, $\\zeta_4 = i$ and the extension is $\\Q(i)$, with the two symmetries $i \\mapsto i$ and $i \\mapsto -i$.\n",
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