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  "body": "\n## Statement\n\nLet $\\varphi$ be a first-order [[sentence|DKFGQTRYZE]] in the signature of rings $\\set{+, -, \\cdot, 0, 1}$.\nThe following are equivalent:\n\n1. $\\varphi$ holds in the field $\\C$ of [[complex numbers|09FTAET5Z4]].\n2. $\\varphi$ holds in every [[algebraically closed field|0PF6TM8TWT]] of [[characteristic|1ZB13N78XW]] $0$.\n3. $\\varphi$ holds in some algebraically closed field of characteristic $0$.\n4. There is an integer $m$ such that $\\varphi$ holds in every algebraically closed field of characteristic $p$, for every prime $p \u003e m$.\n5. $\\varphi$ holds in algebraically closed fields of characteristic $p$ for infinitely many primes $p$.\n\n## Remarks\n\nClauses 1, 2 and 3 are equivalent because the theory $\\mathrm{ACF}_0$ of [[algebraically closed fields|MT8TDVPBXB]] of characteristic $0$ is complete.\nSuppose $\\varphi$ holds in $\\mathrm{ACF}_0$.\nBy the [[compactness theorem|1NR3HAWG9D]], $\\varphi$ follows from $\\mathrm{ACF}$ and finitely many of the axioms $1 + \\cdots + 1 \\ne 0$.\nSo it holds in characteristic $p$ for all large $p$, which is clause 4.\nClause 4 gives 5, since there are infinitely many primes.\nIf $\\varphi$ fails in $\\mathrm{ACF}_0$, then $\\neg \\varphi$ holds there, so $\\neg \\varphi$ holds in all large characteristics and 5 fails.\n\nExample: fix $n$ and $d$.\nThe statement \"every injective polynomial map $K^n \\to K^n$ whose components have degree at most $d$ is surjective\" is a first-order sentence about the field $K$.\nIt holds in the algebraic closure of each finite field $\\F_p$, so by clause 5 it holds in $\\C$.\nThis is the model-theoretic proof of the Ax–Grothendieck theorem.\n\nThe principle holds only for first-order sentences.\nThe statement \"every element is algebraic over the prime field\" holds in the algebraic closure of each $\\F_p$ but fails in $\\C$.\nIt is not expressible by a single first-order sentence.\n\n## Explanations\n\n### L1 human\n\nAlgebraically closed fields of the same characteristic cannot be told apart by first-order statements in the language of rings.\nSo a statement of this kind about the complex numbers is true exactly when it is true in every algebraically closed field of characteristic $0$.\nIt is also true exactly when it holds in algebraically closed fields of all large prime characteristics.\n\n### L2 human\n\nThe principle lets one prove algebraic facts about $\\C$ by working over finite fields, where counting arguments are available, or the other way around.\nIt comes from the completeness of $\\mathrm{ACF}_p$ for each $p$, which itself follows from quantifier elimination or from uncountable categoricity and the Łoś–Vaught test.\nTake the algebraic closures of the fields $\\F_p$ and a non-principal ultrafilter on the primes.\nTheir ultraproduct is an algebraically closed field of characteristic $0$, which gives another view of clause 4.\n",
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