  We study the model theory of the $2$-sorted structure $(\mathbb{F},
\mathbb{C};\chi)$, where $\mathbb{F}$ is an algebraic closure of a finite field
of characteristic $p$, $\mathbb{C}$ is the field of complex numbers and $\chi:
\mathbb{F} \to \mathbb{C}$ is an injective, multiplication preserving map. We
obtain an axiomatization $\mathrm{ACFC}_p$ of
$\mathrm{Th}(\mathbb{F},\mathbb{C};\chi)$ in a suitable language $L$, classify
the models of $\mathrm{ACFC}_p$ up to isomorphism, prove a modified model
companion result, give various descriptions of definable sets inside a model of
$\mathrm{ACFC}_p$, and deduce that $\mathrm{ACFC}_p$ is $\omega$-stable and has
definability of Morley rank in families.
