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  "body": "\n## Definition\n\nLet $K$ be an [[algebraic number field|MWBDFS2KJ5]] and fix an algebraic closure $\\overline{K}$ of $K$.\nA finite extension $L/K$ is *unramified at every place* if two conditions hold.\nEvery nonzero prime ideal of $\\mathcal{O}_K$ is [[unramified|H5RC67SD24]] in $L$.\nEvery real embedding of $K$ extends only to real embeddings of $L$.\nConsider the finite [[Galois extensions|0QMZAQ6QQ4]] $L/K$ in $\\overline{K}$ with [[abelian|082GHNPJ53]] Galois group that are unramified at every place.\nThe *Hilbert class field* of $K$ is the compositum $H \\subseteq \\overline{K}$ of all these fields $L$.\n\n## Remarks\n\nBy class field theory, $H/K$ is a finite abelian extension, unramified at every place, and the [[Artin map|7938EKMMVR]] gives an isomorphism from the [[ideal class group|WQ46BD3ARY]] $\\mathrm{Cl}(K)$ onto $\\Gal(H/K)$.\nSo $[H : K] = h_K$, and a prime ideal of $\\mathcal{O}_K$ splits completely in $H$ if and only if it is principal.\nEvery ideal of $\\mathcal{O}_K$ becomes principal in $\\mathcal{O}_H$ (the principal ideal theorem).\n\nExample: $K = \\Q(\\sqrt{-5})$ and $L = K(i) = \\Q(\\sqrt{-5}, i)$.\nThe three quadratic subfields of $L$ are $\\Q(i)$, $\\Q(\\sqrt{5})$ and $\\Q(\\sqrt{-5})$, with [[discriminants|E0VZTNPJXA]] $-4$, $5$ and $-20$, so $d_L = (-4) \\cdot 5 \\cdot (-20) = 400 = d_K^2$.\nHence the relative discriminant of $L/K$ has norm $1$ and no finite prime ramifies; $K$ has no real embedding, so $L/K$ is unramified at every place.\nSince $h_K = 2 = [L : K]$, the field $L$ is the Hilbert class field of $K$.\n\nNon-example: $L' = \\Q(\\sqrt{-5}, \\sqrt{2})$ is abelian of degree $2$ over $K$ but not unramified.\nIts quadratic subfields have discriminants $8$, $-20$ and $-40$, so $d_{L'} = 6400 = 16 \\, d_K^2$, and some prime of $K$ above $2$ ramifies in $L'$.\n",
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