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  "body": "\n## Summary\n\nLet $A$ be a [[Krull domain|CB9WJCN9BM]].\nIts divisor class group is the group of formal integer combinations of its height-one primes, modulo the divisors of nonzero fractions.\nIt is trivial exactly when the ring is a [[unique factorisation domain|06YMXJMXYS]], so it measures the failure of unique factorization.\nFor a Dedekind domain it is the ideal class group, and it contains the [[Picard group|DZW58R6T2Q]] of the ring.\n\n## Explanations\n\n### L1 human\n\nIn a ring with unique factorization, every height-one prime ideal is generated by one prime element.\nIn other rings some height-one primes are not principal, and the class group records which combinations of them are principal.\nIn $\\Z[\\sqrt{-5}]$, the prime $P = (2, 1 + \\sqrt{-5})$ is not principal, but $P^2 = (2)$ is.\nSo the class of $P$ is a nonzero element of order $2$; in fact the class group of $\\Z[\\sqrt{-5}]$ has exactly two elements.\n",
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