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  "body": "\n## Definition\n\nLet $p$ be a [[prime number|01826KMG2H]] and let $\\abs{\\cdot}_p$ be the [[p-adic absolute value|WGN82TCEF2]] on the [[rational numbers|0NCJ5WXN6Z]] $\\Q$.\nThe field $\\Q_p$ of *$p$-adic numbers* is the [[completion|06RW4RKWH7]] of $\\Q$ for the metric $d(x, y) = \\abs{x - y}_p$.\nA $p$-adic number is an element of $\\Q_p$.\nThe field operations and the absolute value $\\abs{\\cdot}_p$ extend from $\\Q$ to $\\Q_p$ by continuity.\n\n## Remarks\n\nEvery $x \\in \\Q_p$ with $x \\neq 0$ has a unique expansion $x = \\sum_{n \\ge m} a_n p^n$ with $m \\in \\Z$, digits $a_n \\in \\set{0, 1, \\ldots, p - 1}$ and $a_m \\neq 0$.\nThen $\\abs{x}_p = p^{-m}$.\nExample: $-1 = \\sum_{n \\ge 0} (p - 1) p^n$ in $\\Q_p$.\nCheck: the partial sum up to $p^{N-1}$ is $p^N - 1$, and $\\abs{(p^N - 1) - (-1)}_p = p^{-N}$ tends to $0$.\nNon-example: the formal sum $\\sum_{n \\ge 0} p^{-n}$ is not a $p$-adic number, because its terms have $\\abs{p^{-n}}_p = p^n$ and do not tend to $0$.\nThe elements with $\\abs{x}_p \\le 1$ are the [[p-adic integers|V6TJAFZ4MD]] $\\Z_p$.\nBy [[Ostrowski's theorem|GCQ23XT6VD]], the completions of $\\Q$ are $\\R$ and the fields $\\Q_p$.\n",
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