Theory of Algebraically Closed Fields of Characteristic p is Complete

Theorem
Let $ACF_p$ be the theory of algebraically closed fields of characteristic $p$ in the language $\mathcal L_r = \{0,1,+,-,\cdot\}$ for rings, where $0,1$ are constants and $+,-,\cdot$ are binary functions.

$ACF_p$ 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 $\kappa$.


 * $ACF_p$ is satisfiable, for example:

$\C$ is an algebraically closed field of characteristic $0$, and if $p$ is a prime, then the algebraic closure of $\Z / \Z_p$ is an algebraically closed field of characteristic $p$.


 * $ACF_p$ has no finite models:

This is because all algebraically closed fields are infinite.


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

In fact, it is $\kappa$-categorical for all uncountable $\kappa$. This follows from the theorems that fields of uncountable cardinality $\kappa$ have transcendence degree $\kappa$ and Algebraically Closed Fields are Isomorphic iff they have the same Characteristic and Transcendence Degree.