Theory of Algebraically Closed Fields of Characteristic p is Complete
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Let $p$ be either $0$ or a prime number.
- $0, 1$ are constants
- $+, -, \cdot$ are binary functions.
Then $ACF_p$ is complete.
Thus $ACF_p$ is satisfiable.
No Finite Models
$ACF_p$ has no finite models
- Algebraically Closed Fields are Isomorphic iff they have the same Characteristic and Transcendence Degree
it follows that:
- $ACF_p$ is $\kappa$-categorical for all uncountable $\kappa$
- $ACF_p$ is $\kappa$-categorical for some uncountable $\kappa$
as was to be proved.