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  "body": "\n## Statement\n\nThe compactness theorem implies the following result, stated by Abraham Robinson in his 1949 dissertation.\n\nRobinson's principle: If a first-order sentence holds in every [[field|04C5ZBG1WT]] of [[characteristic|0F6ES3BRYP]] zero, then there exists a constant $p$ such that the sentence holds for every field of characteristic larger than $p.$ This can be seen as follows: suppose $\\varphi$ is a sentence that holds in every field of characteristic zero. Then its negation $\\lnot \\varphi,$ together with the field axioms and the infinite sequence of sentences\n\n$$1 + 1 \\neq 0, \\;\\; 1 + 1 + 1 \\neq 0, \\; \\ldots$$\n\nis not satisfiable (because there is no field of characteristic 0 in which $\\lnot \\varphi$ holds, and the infinite sequence of sentences ensures any model would be a field of characteristic 0). Therefore, there is a finite subset $A$ of these sentences that is not satisfiable. $A$ must contain $\\lnot \\varphi$ because otherwise it would be satisfiable. Because adding more sentences to $A$ does not change unsatisfiability, we can assume that $A$ contains the field axioms and, for some $k,$ the first $k$ sentences of the form $1 + 1 + \\cdots + 1 \\neq 0.$ Let $B$ contain all the sentences of $A$ except $\\lnot \\varphi.$ Then any field with a characteristic greater than $k$ is a model of $B,$ and $\\lnot \\varphi$ together with $B$ is not satisfiable. This means that $\\varphi$ must hold in every model of $B,$ which means precisely that $\\varphi$ holds in every field of characteristic greater than $k.$ This completes the proof.\n\nThe Lefschetz principle, one of the first examples of a transfer principle, extends this result. A first-order sentence $\\varphi$ in the language of [[rings|0B1261B5ED]] is true in some (or equivalently, in every) algebraically closed field of characteristic 0 (such as the [[complex numbers|09FTAET5Z4]] for instance) if and only if there exist infinitely many primes $p$ for which $\\varphi$ is true in some algebraically closed field of characteristic $p,$ in which case $\\varphi$ is true in all algebraically closed fields of sufficiently large non-0 characteristic $p.$ One consequence is the following special case of the Ax–Grothendieck theorem: all injective [[complex|09FTAET5Z4]] polynomials $\\mathbb{C}^n \\to \\mathbb{C}^n$ are surjective (indeed, it can even be shown that its inverse will also be a polynomial). In fact, the surjectivity conclusion remains true for any injective polynomial $F^n \\to F^n$ where $F$ is a finite field or the algebraic closure of such a field.\n",
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