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  "body": "\n## Summary\n\nThe class number $h_K$ of a [[number field|MWBDFS2KJ5]] $K$ is the order of its [[ideal class group|WQ46BD3ARY]].\nIt is always finite, and $h_K = 1$ exactly when the ring of integers $\\mathcal{O}_K$ is a principal ideal domain, equivalently a unique factorisation domain.\nClass numbers appear in the [[class number formula|55XX5A7XZV]], and the question of which fields have a given class number is the [[class number problem|KTGAS9DGRX]].\n\n## Explanations\n\n### L1 human\n\nThe class number counts how many essentially different kinds of ideals a number ring has.\nIf it is $1$, every ideal is generated by one number, and numbers factor into primes in only one way, as in the ordinary integers.\nFor example, the Gaussian integers $\\Z[i]$ have class number $1$.\nThe ring $\\Z[\\sqrt{-5}]$ has class number $2$, and there $6 = 2 \\cdot 3 = (1 + \\sqrt{-5})(1 - \\sqrt{-5})$ factors in two ways.\n",
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