Tame structures via character sums over finite fields

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 $◃$ is any translation-invariant circular order on the multiplicative group ${\mathbb{F}}^{\times }$, then $(\mathbb{F},◃)$ 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.

entirely rewritten, 25 pages, a number of results in the earlier versions removed (results about acl-completeness now incorporated into the more general framework of interpolative fusions, decidability results will be presented in a future paper)

References in corpus (2)

• Generic expansion and Skolemization in NSOP${}_1$ theories
• Interpolative Fusions I