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  "body": "\n## Definition\n\nLet $K$ be a field with an absolute value $\\abs{\\cdot}$, that is, a [[normed field|ZFW9Q3WTAQ]], with the metric $d(x, y) = \\abs{x - y}$.\nA *completion* of $(K, \\abs{\\cdot})$ is a field $\\widehat{K}$ with an absolute value $\\norm{\\cdot}$, together with a field homomorphism $\\iota \\colon K \\to \\widehat{K}$, such that:\n\n1. $\\norm{\\iota(x)} = \\abs{x}$ for all $x \\in K$.\n2. $\\widehat{K}$ is a [[complete metric space|08EWT7J0SC]] for the metric $(a, b) \\mapsto \\norm{a - b}$.\n3. The image $\\iota(K)$ is dense in $\\widehat{K}$.\n\n## Remarks\n\nA completion exists and is unique up to a unique isomorphism of fields that preserves the absolute values and commutes with the maps from $K$.\nOne construction: the [[Cauchy sequences|0QZMTN5AXX]] in $K$ form a ring under termwise operations.\nThe sequences tending to $0$ form a maximal ideal, and the quotient is a field.\nThe absolute value of the class of $(x_n)$ is $\\lim_n \\abs{x_n}$.\nAs a metric space it is the [[completion of the metric space|0G4J7Z7WPJ]] $K$.\nExample: $\\R$ with the usual absolute value and the inclusion $\\Q \\to \\R$ is a completion of $\\Q$; clause 3 holds because every real number is a limit of rationals.\nExample: for a prime $p$, the [[p-adic numbers|BT6T702C28]] $\\Q_p$ are the completion of $\\Q$ for the [[p-adic absolute value|WGN82TCEF2]].\nNon-example: the inclusion $\\Q \\to \\C$ satisfies clauses 1 and 2 for the usual absolute values but fails clause 3, since the closure of $\\Q$ in $\\C$ is $\\R$.\nFor a [[non-Archimedean absolute value|1H5F1BP6XS]], the completion has the same value group and the same residue field as $K$.\n",
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