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  "body": "\n## Definition\n\nLet $A$ be an [[integral domain|09S3PFBEPM]] with [[field of fractions|0VF11ZTTMD]] $K$.\nA *discrete valuation* of $K$ is a surjective map $v \\colon K^\\times \\to \\Z$ with $v(xy) = v(x) + v(y)$ and $v(x + y) \\ge \\min(v(x), v(y))$ whenever $x, y, x + y \\in K^\\times$; its ring is $R_v = \\set{0} \\cup \\set{x \\in K^\\times : v(x) \\ge 0}$, a [[discrete valuation ring|MGM3CNJW9V]].\n$A$ is a *Krull domain* if there is a family $(v_i)_{i \\in I}$ of discrete valuations of $K$ such that:\n\n1. $A = \\bigcap_{i \\in I} R_{v_i}$;\n2. for every $x \\in K^\\times$, $v_i(x) \\ne 0$ for only finitely many $i \\in I$.\n\n## Remarks\n\nExample: $\\Z$ is a Krull domain, with the $p$-adic valuations $v_p$ of $\\Q$ for all primes $p$.\nA rational number $x$ lies in $\\Z$ exactly when $v_p(x) \\ge 0$ for all $p$, and a nonzero rational number involves only finitely many primes.\nMore generally every [[unique factorisation domain|06YMXJMXYS]] and every Noetherian [[integrally closed domain|H20P600HQS]] is a Krull domain.\nNon-example: $\\Z[\\sqrt{-3}]$ is not a Krull domain.\nAn intersection of valuation rings is integrally closed, but $(1 + \\sqrt{-3})/2$ is a root of $x^2 - x + 1$ and does not lie in $\\Z[\\sqrt{-3}]$.\nSo clause 1 fails for every family.\nThe family can be chosen as the valuations of the localizations $A_P$ at the height-one primes $P$ of $A$.\nBy the [[Mori–Nagata theorem|J8YYYP5CDN]], the integral closure of a Noetherian domain is a Krull domain.\n",
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