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  "body": "\n## Statement\n\nLet $L$ be a [[finite extension|0C1Y04SZ1W]] of $\\Q$ that is a [[Galois extension|0QMZAQ6QQ4]] with [[abelian|082GHNPJ53]] Galois group.\nThen there is a positive integer $n$ such that $L$ is contained in the [[cyclotomic field|NN77VYSGXR]] $\\Q(\\zeta_n)$.\nHere $\\zeta_n$ is a primitive $n$-th [[root of unity|0YV2CKP2AV]].\n\n## Remarks\n\nThe converse holds as well: $\\Gal(\\Q(\\zeta_n)/\\Q) \\cong (\\Z/n\\Z)^\\times$ is abelian, so every subfield of $\\Q(\\zeta_n)$ is abelian over $\\Q$.\nOne standard proof reduces to cyclic extensions of prime power degree and studies their ramification groups at the ramified primes.\nA key input is that $\\Q$ has no nontrivial extension unramified at all primes, a consequence of the [[Hermite–Minkowski theorem|KMVW45B0TT]].\nAnother proof comes from [[class field theory|BEKCEYHCHX]]: the ray class field of $\\Q$ for the modulus $n\\infty$ is $\\Q(\\zeta_n)$.\n\nExample: every [[quadratic field|7SV4WA92SM]] is abelian over $\\Q$, so it lies in a cyclotomic field.\nFor instance $\\sqrt{2} = \\zeta_8 + \\zeta_8^{-1}$ lies in $\\Q(\\zeta_8)$, and $\\sqrt{5} = \\zeta_5 - \\zeta_5^2 - \\zeta_5^3 + \\zeta_5^4$ lies in $\\Q(\\zeta_5)$.\n\nThe abelian hypothesis is needed.\nThe field $\\Q(\\sqrt[3]{2})$ is not contained in any $\\Q(\\zeta_n)$: it is not Galois over $\\Q$, while every subfield of $\\Q(\\zeta_n)$ is Galois over $\\Q$.\n\n## Explanations\n\n### L1 human\n\nRoots of unity are the solutions of $x^n = 1$.\nThe theorem says that every number field with a commutative Galois group over $\\Q$ can be built from roots of unity.\nSo the abelian extensions of $\\Q$ have a complete and explicit description.\n\n### L2 human\n\nThe maximal abelian extension of $\\Q$ is the field generated by all roots of unity.\nIts Galois group is $\\widehat{\\Z}^\\times$, the unit group of the profinite completion $\\widehat{\\Z}$ of $\\Z$.\nHilbert's twelfth problem asks for similar explicit generators for the abelian extensions of other number fields.\nFor imaginary quadratic fields this is achieved by [[complex multiplication|D3HEJTVY7G]], using values of elliptic and modular functions.\n",
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