Tame structures via character sums over finite fields

TL;DR

This paper establishes a model companion for algebraically closed fields with multiplicative circular orders, using character sums over finite fields to analyze their structure and properties.

Contribution

It introduces the theory ACFO for algebraically closed fields with circular orders and demonstrates that certain finite field structures are models of this theory.

Findings

ACFO is a model companion for algebraically closed fields with circular orders.

Finite fields with translation-invariant circular orders are models of ACFO.

The approach uses number-theoretic character sums to analyze these structures.

Abstract

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.