Theory of Algebraically Closed Fields of Characteristic p is Complete
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Theorem
Let $p$ be either $0$ or a prime number.
Let $ACF_p$ be the theory of algebraically closed fields of characteristic $p$ in the language $\LL_r = \set {0, 1, +, -, \cdot}$ for rings, where:
- $0, 1$ are constants
and:
- $+, -, \cdot$ are binary functions.
Then $ACF_p$ is complete.
Proof
By the Łoś-Vaught Test, it suffices to show that $ACF_p$ is satisfiable, has no finite models, and is $\kappa$-categorical for some uncountable $\kappa$.
Satisfiability
$\C$ is an algebraically closed field of characteristic $0$.
If $p$ is a prime, then the algebraic closure of $\Z / \Z_p$ is an algebraically closed field of characteristic $p$.
Thus $ACF_p$ is satisfiable.
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$\Box$
No Finite Models
From Algebraically Closed Field is Infinite:
$ACF_p$ has no finite models
$\Box$
$\kappa$-Categorical
From:
and:
it follows that:
- $ACF_p$ is $\kappa$-categorical for all uncountable $\kappa$
Hence:
- $ACF_p$ is $\kappa$-categorical for some uncountable $\kappa$
as was to be proved.
$\blacksquare$