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  "body": "\n## Summary\n\nThe $p$-adic integers are the [[p-adic numbers|BT6T702C28]] $x$ with $\\abs{x}_p \\le 1$.\nThey form a ring $\\Z_p$, which is a [[discrete valuation ring|MGM3CNJW9V]] with maximal ideal $p\\Z_p$ and residue field $\\F_p$.\nThe ring $\\Z_p$ is compact, and it is the inverse limit of the rings $\\Z / p^n \\Z$.\nIt holds, in one object, the information of all congruences modulo powers of $p$.\n\n## Explanations\n\n### L1 human\n\nA $p$-adic integer is a base $p$ expansion $a_0 + a_1 p + a_2 p^2 + \\cdots$ that goes on forever to the left.\nCutting it after $n$ digits gives a residue modulo $p^n$, and these residues fit together.\nEvery ordinary integer is a $p$-adic integer, and so is every fraction whose denominator is not divisible by $p$.\nFor example, $1/2$ is a $3$-adic integer, but $1/3$ is not.\n",
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