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  "body": "\n## Definition\n\nLet $K$ be a field with a [[valuation|B85T5VKR3M]] $v$.\nLet $\\mathcal{O} = \\set{x \\in K : v(x) \\ge 0}$ be its [[valuation ring|QSX12D8RQM]] and $\\mathfrak{m} = \\set{x \\in K : v(x) \u003e 0}$ its [[maximal ideal|09J6KGT65G]].\nThe valued field $(K, v)$ is *Henselian* if the following holds.\nFor every monic $f \\in \\mathcal{O}[X]$ and every $a \\in \\mathcal{O}$ with $f(a) \\in \\mathfrak{m}$ and $f'(a) \\notin \\mathfrak{m}$, there is $b \\in \\mathcal{O}$ with $f(b) = 0$ and $b - a \\in \\mathfrak{m}$.\n\nHere $f'$ is the formal [[derivative|0AD4VS02H3]] of $f$.\n\n## Remarks\n\nExample: for a prime $p$, the field $\\Q_p$ with the $p$-adic valuation is Henselian; this is [[Hensel's lemma|010QVB71T2]] in its $p$-adic form.\nMore generally, a field that is complete for a valuation with values in $\\R$ is Henselian.\nNon-example: $\\Q$ with the $7$-adic valuation $v_7$ is not Henselian.\nTake $f = X^2 - 2$ and $a = 3$: $f(3) = 7 \\in \\mathfrak{m}$ and $f'(3) = 6 \\notin \\mathfrak{m}$, but $f$ has no root in $\\Q$.\nEquivalent forms: $(K, v)$ is Henselian if and only if $\\mathcal{O}$ is a [[Henselian ring|RNXPV3KC5J]].\nIt is Henselian if and only if $v$ extends to a unique valuation on each algebraic extension of $K$.\nThe element $b$ is then unique.\nEvery field with the trivial valuation is Henselian.\n",
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