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  "body": "\n## Statement\n\nLet $K$ be a field that is complete for a [[non-Archimedean absolute value|1H5F1BP6XS]] $\\abs{\\cdot}$.\nLet $\\overline{K}$ be an [[algebraic closure|0MCXX29HWY]] of $K$, with the unique extension of $\\abs{\\cdot}$ to $\\overline{K}$.\nLet $\\alpha \\in \\overline{K}$ be [[separable|0N019F6RE9]] over $K$, and let $\\alpha = \\alpha_1, \\alpha_2, \\ldots, \\alpha_n$ be the distinct conjugates of $\\alpha$ over $K$, with $n \\ge 2$.\nLet $\\beta \\in \\overline{K}$ satisfy\n$$\\abs{\\beta - \\alpha} \u003c \\abs{\\alpha - \\alpha_i} \\quad \\text{for } i = 2, \\ldots, n.$$\nThen $K(\\alpha) \\subseteq K(\\beta)$.\n\n## Remarks\n\nProof idea: let $\\sigma$ be an automorphism of $\\overline{K}$ that fixes $K(\\beta)$.\nBy uniqueness of the extended absolute value, $\\abs{\\sigma(x)} = \\abs{x}$ for all $x$, so $\\abs{\\sigma(\\alpha) - \\beta} = \\abs{\\sigma(\\alpha - \\beta)} = \\abs{\\alpha - \\beta}$.\nThe strong triangle inequality gives $\\abs{\\sigma(\\alpha) - \\alpha} \\le \\max(\\abs{\\sigma(\\alpha) - \\beta}, \\abs{\\beta - \\alpha}) \u003c \\abs{\\alpha - \\alpha_i}$ for $i \\ge 2$, so $\\sigma(\\alpha) = \\alpha$.\nAs $\\alpha$ is separable and fixed by all such $\\sigma$, it lies in $K(\\beta)$.\nThe same proof works for a Henselian valued field, since there too the absolute value extends uniquely.\nFor $n = 1$ the element $\\alpha$ lies in $K$, so the conclusion holds for every $\\beta$.\n\nApplication: let $f$ be a monic irreducible separable polynomial over $K$, and let $g$ be a monic polynomial over $K$ of the same degree with close enough coefficients.\nThen $g$ is irreducible, and each root of $g$ generates the same extension of $K$ as a suitable root of $f$.\nThis shows that every finite extension of $\\Q_p$ is generated by a root of a polynomial with rational coefficients.\nIt is also the key step in the proof that the [[complex p-adic numbers|FKF9YGBKHF]] form an algebraically closed field.\n\n## Explanations\n\n### L1 human\n\nIn a $p$-adic field, let $\\beta$ be closer to $\\alpha$ than any other conjugate of $\\alpha$ is.\nThen $\\beta$ already \"knows\" $\\alpha$: the field made from $\\beta$ contains $\\alpha$.\nCompleteness is needed.\nIn $\\Q$ with the $7$-adic absolute value, rational numbers can be as close as one likes to a square root of $2$.\nYet they generate only $\\Q$, which does not contain that square root.\n\n### L2 human\n\nKrasner's lemma is the basic continuity statement for roots in local fields.\nWith the [[Newton polygon|H7QNCE2QZJ]] and [[Hensel's lemma|010QVB71T2]] it is one of the three main tools for studying extensions of $\\Q_p$.\nIt implies that a finite separable extension of $\\Q_p$ does not change when the defining polynomial is perturbed slightly.\nWith a compactness argument, this shows that $\\Q_p$ has only finitely many extensions of each degree.\nIt also gives that the completion of an algebraic closure of $\\Q_p$ is again algebraically closed.\n",
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