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  "body": "\n## Summary\n\nThe Hilbert class field of a number field $K$ is the largest abelian extension of $K$ that is unramified at every place.\nBy [[class field theory|BEKCEYHCHX]], its Galois group over $K$ is isomorphic to the [[ideal class group|WQ46BD3ARY]] of $K$, so its degree is the [[class number|KTKK9B1X41]].\nA prime ideal of $K$ splits completely in it exactly when it is principal.\nSo the class group, an arithmetic object, is realised as a Galois group.\n\n## Explanations\n\n### L1 human\n\nThe class group measures how far the ring of integers of a field is from having unique factorisation.\nThe Hilbert class field is a bigger field that makes this measure visible as a group of symmetries.\nFor $K = \\Q(\\sqrt{-5})$, with class number $2$, the Hilbert class field is $K(i) = \\Q(\\sqrt{-5}, i)$, an extension of degree $2$.\nEvery ideal of $K$ becomes principal in the Hilbert class field: for example $(2, 1 + \\sqrt{-5})$ is generated there by $1 + i$.\n",
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