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I once read (I think) the following equivalent formulation of the Minor Lefschetz principle:

  • If an elementary sentence holds for one algebraically closed field, then it holds for every algebraically closed field.

So without restrictions on the characteristic.

Is this version correct, is it really equivalent ? (I can not find any source anymore.)

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    $\begingroup$ If "elementary" means first order, this is false: $\forall x,2x=0$ holds for $\overline{\Bbb{F}}_2$, but not for $\mathbb{C}$. $\endgroup$
    – abx
    Sep 16, 2022 at 16:02

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The correct statement is that the following are equivalent, for a sentence $\varphi$ in the first-order language of fields.

  1. $\varphi$ is true in some algebraically closed field of characteristic $0$.
  2. $\varphi$ is true in every algebraically closed field of characteristic $0$.
  3. There exist arbitrarily large $p$ such that $\varphi$ is true in some algebraically closed field of characteristic $p$.
  4. For all sufficiently large $p$, $\varphi$ is true in every algebraically closed field of characteristic $p$.

The proof is almost immediate from compactness and the completeness of the theories of algebraically closed fields of fixed characteristic (which is itself an easy consequence of quantifier-elimination).

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