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  "body": "\n## Summary\n\nA Henselian field is a [[valued field|B85T5VKR3M]] in which the conclusion of [[Hensel's lemma|010QVB71T2]] holds, see the [[definition|Z3KPTM2YWX]].\nIn it, a simple root of a polynomial modulo the maximal ideal lifts to a true root.\nIts [[valuation ring|QSX12D8RQM]] is a [[Henselian ring|RNXPV3KC5J]].\nComplete fields for a discrete valuation, such as the [[p-adic numbers|BT6T702C28]] and [[fields of formal Laurent series|YZWYF7DXTW]], are Henselian.\nIn a Henselian field the valuation extends in only one way to each algebraic extension, which makes their extension theory clean.\n\n## Explanations\n\n### L1 human\n\nOver the $7$-adic numbers, one can solve $x^2 = 2$ by first solving it modulo $7$ and then correcting the answer step by step.\nModulo $7$, the number $3$ works, since $9 - 2 = 7$.\nA field is Henselian when this \"solve approximately, then refine\" method always works for simple roots.\nThe rationals with the $7$-adic valuation are not Henselian: the approximate root $3$ exists, but $\\sqrt 2$ is not rational.\n",
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