Theory of Algebraically Closed Fields of Characteristic p is Complete

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.

No Finite Models
From Algebraically Closed Field is Infinite:

$ACF_p$ has no finite models

$\kappa$-Categorical
From:
 * Field of Uncountable Cardinality $\kappa$ has Transcendence Degree $\kappa$

and:
 * 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 $\kappa$

Hence:
 * $ACF_p$ is $\kappa$-categorical for some uncountable $\kappa$

as was to be proved.