Model theory of finite-by-Presburger Abelian groups and finite extensions of p-adic fields

2016/03/29 by Jamshid Derakhshan, Angus Macintyre, Derakhshan, Jamshid +1
Mathematics · #03C10 #03C60 #03C65 #11U09 #12E30 #12J12 #12J20 #12J25 #Advanced Topology and Set Theory #Algebraic Geometry and Number Theory #FOS: Mathematics #Logic (math.LO) #advanced mathematical theories #math.LO #msc:03C10 #msc:03C60 #msc:03C65 #msc:11U09 #msc:12E30 #msc:12J12 #msc:12J20 #msc:12J25

paper · pdf · doi:10.48550/arxiv.1603.08601

arxiv created 2016/03/29 · openalex publication_date 2016/03/29 · arxiv updated 2016/03/30 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

We define a class of pre-ordered abelian groups that we call finite-by-Presburger groups, and prove that their theory is model-complete. We show that certain quotients of the multiplicative group of a local field of characteristic zero are finite-by-Presburger and interpret the higher residue rings of the local field. We apply these results to give a new proof of the model completeness in the ring language of a local field of characteristic zero (a result that follows also from work of Prestel-Roquette).

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