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  "body": "\n## Statement\n\nLet $A$ be a [[Noetherian|0MVH4Y1CRZ]] [[integral domain|09S3PFBEPM]] with [[field of fractions|0VF11ZTTMD]] $K$.\nThen the [[integral closure|G0V86SJBQW]] of $A$ in $K$ is a [[Krull domain|CB9WJCN9BM]].\n\n## Remarks\n\nA Noetherian integrally closed domain is a Krull domain, so the theorem says that integral closure keeps part of the Noetherian property, even when it loses the rest.\nThe integral closure of a Noetherian domain need not be Noetherian, and Nagata gave such examples.\nSo the class of Krull domains is the natural home for integral closures of Noetherian domains.\nIn dimension $1$, the [[Krull–Akizuki theorem|FSXA3CF365]] gives more: the integral closure is Noetherian.\nFor example, the integral closure of $\\Z[\\sqrt{-3}]$ in $\\Q(\\sqrt{-3})$ is $\\Z[(1 + \\sqrt{-3})/2]$, a Dedekind domain and so a Krull domain.\n\n## Explanations\n\n### L1 human\n\nMaking a ring integrally closed can destroy the Noetherian property.\nThe theorem says that something useful survives.\nThe closed ring is a Krull domain, which means that it still has a good theory of divisors, as in a ring with unique factorisation up to a class group.\n\n### L2 human\n\nKrull domains are the domains that are intersections of discrete valuation rings with a finiteness condition, and they carry a divisor class group.\nThe theorem lets one use divisor theory on the normalisation of any Noetherian domain, for example on the normalisation of the coordinate ring of a singular affine variety.\nCompare the [[Krull–Akizuki theorem|FSXA3CF365]], which in dimension one gives a Noetherian normalisation.\n",
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