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  "body": "\n## Summary\n\nThis part of model [[theory|92SWXST1ZM]] studies the first-order theory $\\mathrm{ACF}$ of [[algebraically closed fields|0PF6TM8TWT]] in the language of rings.\nIt also studies the completions $\\mathrm{ACF}_p$, one for each characteristic $p$.\nThe theory $\\mathrm{ACF}$ admits [[quantifier elimination|7FDDXV3JK7]].\nEach $\\mathrm{ACF}_p$ is [[complete|655K8Y7VS8]] and decidable, and categorical in every uncountable cardinality.\nThese facts give the [[Lefschetz principle|SKTMN14X5P]] and short proofs of results in algebra, such as the [[Ax–Grothendieck theorem|M2XAN2HGZH]].\nIt is also the basic example of a [[strongly minimal theory|PNGY60EBDJ]].\n\n## Explanations\n\n### L1 human\n\nModel theory asks which statements about a mathematical structure can be written with \"for all\", \"there exists\", and the symbols $+$, $\\cdot$, $0$, $1$, $=$.\nFor algebraically closed fields, every such statement is equivalent to a statement without \"for all\" and \"there exists\", that is, to a combination of polynomial equations and inequations.\nFor example, \"there is $x$ with $a x^2 + b x + c = 0$\" is equivalent to \"$a \\ne 0$, or $b \\ne 0$, or $c = 0$\".\nAs a consequence, a statement of this kind that holds in the complex numbers holds in every algebraically closed field of characteristic $0$.\nThis lets one move results from $\\C$ to other fields and back.\n",
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