{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "P32FQE1H0D"
    }
  ],
  "body": "\n## Summary\n\nA field $K$ is pseudo algebraically closed, or PAC, if every absolutely irreducible algebraic variety defined over $K$ has a point with coordinates in $K$.\n[[Algebraically closed fields|0PF6TM8TWT]] are PAC, but many PAC fields are far from algebraically closed.\nPAC fields are a central notion of field arithmetic, and their theory links Galois theory with model theory through the [[absolute Galois group|PGKWBJDTKX]].\n\n## Explanations\n\n### L1 human\n\nAn algebraically closed field contains a root of every polynomial equation in one unknown.\nA PAC field asks for something different.\nEvery system of polynomial equations in several unknowns whose solution set is geometrically \"one piece\" must have a solution in the field.\nThe field of rational numbers is not PAC, since $x^2 + y^2 = -1$ has no rational solution although its solution set over $\\C$ is one irreducible curve.\nExamples of PAC fields include the infinite algebraic extensions of a finite field.\n",
  "commit": "5fc425f97f1f6b11da5892c2948ef8c5d8805665",
  "front": {
    "kind": "concept",
    "msc": [
      "12E30",
      "12L12",
      "14G05"
    ],
    "name": "Pseudo algebraically closed field",
    "same_as": {
      "wikidata": "Q7254461",
      "wikipedia": "Pseudo algebraically closed field"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Pseudo_algebraically_closed_field"
      }
    ],
    "tag": "92SQ10PN6J"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/92SQ10PN6J/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/92SQ10PN6J.md",
  "msc": [
    "12E30",
    "12L12",
    "14G05"
  ],
  "name": "Pseudo algebraically closed field",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/92S/92SQ10PN6J.md",
  "status": null,
  "tag": "92SQ10PN6J",
  "uri": "m2://92SQ10PN6J",
  "url": "https://math2.tamnd.com/wiki/pseudo-algebraically-closed-field/",
  "writing": {
    "tag": "92SQ10PN6J",
    "kind": "concept",
    "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
      }
    ]
  }
}