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  "body": "\n## Summary\n\nAn integral scheme is a nonempty [[scheme|5A35TYZD02]] whose ring of functions on each nonempty open set is an [[integral domain|0QVD7WBFGN]].\nEquivalently, it is [[reduced|ZXSN8D4MCB]] and its underlying space is irreducible.\nIntegral schemes are the scheme version of varieties, and each has a function field: the local ring at its [[generic point|3A19C4YXGE]].\n\n## Explanations\n\n### L1 human\n\nAn integral scheme is \"all in one piece\" and has no nilpotent fuzz.\nThe line $\\operatorname{Spec} k[x]$ and $\\operatorname{Spec} \\Z$ are integral.\nThe union of the two coordinate axes, $\\operatorname{Spec} k[x, y]/(xy)$, is reduced but made of two pieces, so it is not integral: $x \\cdot y = 0$ with $x \\ne 0$ and $y \\ne 0$.\nThe double point $\\operatorname{Spec} k[x]/(x^2)$ is one piece but not reduced, so it is not integral either.\n",
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