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  "body": "\n## Definition\n\nLet $A$ be an [[integral domain|0QVD7WBFGN]] with [[field of fractions|0F2PTGX1A2]] $K$.\nThe domain $A$ is *integrally closed* if every element of $K$ that is [[integral|1EHN4KPSFS]] over $A$ lies in $A$.\nEquivalently, $A$ is equal to its [[integral closure|G0V86SJBQW]] in $K$.\n\n## Remarks\n\nExample: $\\Z$ is integrally closed.\nIf $p/q$ in lowest terms is a root of $x^n + a_{n-1} x^{n-1} + \\cdots + a_0$ with $a_i \\in \\Z$, then $p^n = -q(a_{n-1} p^{n-1} + \\cdots + a_0 q^{n-1})$, so $q$ divides $p^n$, which forces $q = \\pm 1$.\nThe same argument shows that every [[unique factorisation domain|0W4X9G6263]] is integrally closed.\n\nNon-example: $\\Z[\\sqrt{-3}]$ is not integrally closed.\nThe element $(1 + \\sqrt{-3})/2$ of its field of fractions is a root of $x^2 - x + 1$, but it is not in $\\Z[\\sqrt{-3}]$.\n\nAn integrally closed domain is also called a normal domain.\nA domain is integrally closed if and only if all its localizations at prime ideals are integrally closed.\n",
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