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  "body": "\n## Summary\n\nA Henselian ring is a [[local ring|MZ02BKPHJA]] in which [[Hensel's lemma|010QVB71T2]] holds.\nThere, factorizations of polynomials over the [[residue field|NJ3APQ8BHH]] into coprime factors lift to the ring.\nEvery complete local ring, such as the $p$-adic integers or a formal power series ring, is Henselian.\nHenselian rings behave like small neighbourhoods of a point in the étale topology.\nEvery local ring has a [[Henselization|917PGHVZPM]], a smallest Henselian ring over it.\n\n## Explanations\n\n### L1 human\n\nHensel's lemma says that a root of a polynomial modulo $p$ can be refined to a root modulo $p^2$, $p^3$, and so on, if the root is simple.\nA Henselian ring is a local ring where this refinement always ends in an actual root inside the ring.\nThe $p$-adic integers have this property, but the fractions with denominators prime to $p$ do not.\nFor example, $x^2 + 1$ has a simple root modulo $5$, yet no square root of $-1$ is rational.\nThe [[definition|WSQ8GH8XWZ]] states the lifting condition.\n",
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