  We show that the theory of algebraically closed fields with multiplicative
circular orders has a model companion $\mathrm{ACFO}$. Using number-theoretic
results on character sums over finite fields, we show that if $\mathbb{F}$ is
an algebraic closure of a finite field, and $\triangleleft$ is any
translation-invariant circular order on the multiplicative group
$\mathbb{F}^\times$, then $(\mathbb{F}, \triangleleft)$ is a model of
$\mathrm{ACFO}$. Our results can be regarded as analogues of Ax's results in
[1] which utilize counting points over finite fields.
