Tame structures via character sums over finite fields

2017/04/12 by Tran, Chieu-Minh
#03B25 #03C10 #03C64 #03C65 #11T24 #12L12 #FOS: Mathematics #Logic (math.LO)

paper · doi:10.48550/arxiv.1704.03853

Abstract

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

Related

• Algebraically Closed Fields with a Generic Multiplicative Character
• Model theory of difference fields with an additive character on the fixed field
• Ax's theorem with an additive character
• Pseudofinite fields with additive and multiplicative character
• Finite field models in additive combinatorics