Local symbols and a first-order definition of the polynomial ring over an ultra-finite field in its fraction field

2023/09/06 by Nguyen, Dong Quan Ngoc
#FOS: Mathematics #Logic (math.LO) #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2309.02738

Abstract

In this paper, we prove the existence of a first-order definition of the polynomial ring over a nonprincipal ultraproduct of finite fields of unbounded cardinalities in its fraction field by a universal-existential formula in the language of rings augmented by an additional constant symbol t. As a consequence, we prove that the full first-order theory of the rational function field over a nonprincipal ultraproduct of finite fields of characteristic 0 is undecidable.

Related

• Uniformly defining valuation rings in Henselian valued fields with finite or pseudo-finite residue fields
• Higher reciprocity law and An analogue of the Grunwald--Wang theorem for the ring of polynomials over an ultra-finite field
• Ultra-Galois theory and an analogue of the Kronecker--Weber theorem for rational function fields over ultra-finite fields
• A universal first order formula defining the ring of integers in a number field
• Nonstandard polynomials: algebraic properties and elementary equivalence