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  "body": "\n## Summary\n\nAn integrally closed domain, or normal domain, is an [[integral domain|0QVD7WBFGN]] that contains every element of its [[field of fractions|0F2PTGX1A2]] that is [[integral|1EHN4KPSFS]] over it.\nEvery [[unique factorisation domain|0W4X9G6263]] is integrally closed, and so is every Dedekind domain and every valuation ring.\nThe [[going-down theorem|YE52MQ13MV]] holds over integrally closed domains.\nIn geometry, integrally closed domains are the coordinate rings of normal varieties, whose singular points form a set of codimension at least two.\n\n## Explanations\n\n### L1 human\n\nA fraction that satisfies a monic equation with coefficients in the ring should already be in the ring.\nFor the integers this is the familiar fact that a rational root of $x^n + a_{n-1} x^{n-1} + \\cdots + a_0$ with integer coefficients is an integer.\nThe ring $k[t^2, t^3]$ of polynomials without a linear term fails this test.\nThe fraction $t = t^3 / t^2$ is a root of $x^2 - t^2$, a monic polynomial over the ring, but $t$ itself is not in the ring.\nGeometrically this ring describes the curve $y^2 = x^3$, which has a cusp at the origin.\n",
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