{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Description\n\nLet $p$ be a prime and let $\\Q_p$ be the field of [[p-adic numbers|BT6T702C28]] with the $p$-adic absolute value $\\abs{\\cdot}_p$.\nLet $\\overline{\\Q}_p$ be an [[algebraic closure|0MCXX29HWY]] of $\\Q_p$.\nBecause $\\Q_p$ is complete, $\\abs{\\cdot}_p$ extends in exactly one way to an absolute value on $\\overline{\\Q}_p$, and this extension is a [[non-Archimedean absolute value|1H5F1BP6XS]].\nThe field $\\overline{\\Q}_p$ is not complete for it.\nThe field $\\C_p$ of *complex $p$-adic numbers* is the [[completion|KFYW0WN9AD]] of $\\overline{\\Q}_p$ for this absolute value.\n\nThe field $\\C_p$ is an [[algebraically closed field|0PF6TM8TWT]] and it is complete.\nThe algebraic closedness follows from [[Krasner's lemma|G8PGN0WQHW]].\nOne approximates a monic polynomial over $\\C_p$ by monic polynomials over $\\overline{\\Q}_p$, whose roots lie in $\\overline{\\Q}_p$.\nThe roots of these approximations can be chosen to form a Cauchy sequence, and its limit in $\\C_p$ is a root of the original polynomial.\n\n## Remarks\n\nThe set of values $\\abs{x}_p$ for $x \\in \\C_p^\\times$ is $p^{\\Q} = \\set{p^r : r \\in \\Q}$, the same as for $\\overline{\\Q}_p$.\nThe residue field of $\\C_p$ is an algebraic closure of the finite field $\\F_p$.\nThe field $\\C_p$ plays for $\\Q_p$ the role that $\\C$ plays for $\\R$, but $\\C_p$ has infinite degree over $\\Q_p$.\n\n## Explanations\n\n### L1 human\n\nTo do analysis with $p$-adic numbers one wants a field where every polynomial has a root and every Cauchy sequence has a limit.\nAdding all roots of polynomials to $\\Q_p$ gives algebraic closedness but loses completeness.\nCompleting once more gives $\\C_p$, and the surprise is that no new polynomial roots are needed after that.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "instance_of": [
      "0PF6TM8TWT"
    ],
    "kind": "object",
    "msc": [
      "12J25",
      "11S80"
    ],
    "name": "Complex p-adic numbers",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/P-adic_number"
      }
    ],
    "tag": "FKF9YGBKHF",
    "uses": [
      "BT6T702C28",
      "0MCXX29HWY",
      "KFYW0WN9AD",
      "1H5F1BP6XS"
    ]
  },
  "kind": "object",
  "link": "https://math2.tamnd.com/node/FKF9YGBKHF/",
  "links": [
    {
      "field": "instance_of",
      "target": "0PF6TM8TWT"
    },
    {
      "field": "uses",
      "target": "BT6T702C28"
    },
    {
      "field": "uses",
      "target": "0MCXX29HWY"
    },
    {
      "field": "uses",
      "target": "KFYW0WN9AD"
    },
    {
      "field": "uses",
      "target": "1H5F1BP6XS"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/FKF9YGBKHF.md",
  "msc": [
    "12J25",
    "11S80"
  ],
  "name": "Complex p-adic numbers",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/FKF/FKF9YGBKHF.md",
  "status": null,
  "tag": "FKF9YGBKHF",
  "uri": "m2://FKF9YGBKHF",
  "url": "https://math2.tamnd.com/wiki/complex-p-adic-numbers/",
  "writing": {
    "tag": "FKF9YGBKHF",
    "kind": "object",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      },
      {
        "what": "field same_as",
        "note": "the identifiers in the databases",
        "have": 0,
        "want": 1
      }
    ]
  }
}