= Characteristic-zero ultraproducts of prime fields
{title2=$\operatorname{char}(\prod_p\mathbb F_p/\mathcal U)=0$}
For a <nonprincipal ultrafilter> on the primes, the <ultraproduct> of the prime fields $\mathbb F_p$ is a field of characteristic zero. Each fixed sentence $n\cdot1\ne0$, $n\geq1$, fails at only finitely many prime divisors of $n$; its cofinite truth set belongs to the ultrafilter, so <Łoś theorem> transfers it to the product. Every sentence eventually true in the prime fields also transfers. However, the sentence asserting that minus one is a square splits along the two infinite prime classes modulo four. Ultrafilters concentrating on these two classes yield different characteristic-zero fields satisfying the same eventual-truth theory, showing that this common theory is incomplete.
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