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  "body": "\n## Summary\n\nThe Henselization of a [[local ring|MZ02BKPHJA]] $R$ is the smallest [[Henselian local ring|RNXPV3KC5J]] that receives a local map from $R$.\nIt is characterized by a universal property among local homomorphisms from $R$ to Henselian local rings.\nIt keeps the residue field of $R$ and, for a Noetherian $R$, it has the same completion as $R$.\nIt is a substitute for the completion that stays algebraic over $R$, and it is used in étale cohomology and in approximation theorems.\n\n## Explanations\n\n### L1 human\n\n[[Hensel's lemma|010QVB71T2]] lets one lift a simple root of a polynomial modulo the maximal ideal to a true root.\nMany local rings do not have this property, and the Henselization adds just enough elements to get it.\nThe completion also has the property, but it adds far more elements, most of them transcendental.\nFor the ring $\\Z_{(p)}$ of fractions with denominator prime to $p$, the Henselization is the ring of $p$-adic integers that are algebraic over $\\Q$.\nSo it lies between $\\Z_{(p)}$ and the full ring of $p$-adic integers.\n",
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