Substructures and uniform elimination for p-adic fields
type: Item
instance of: scholarly article
title: Substructures and uniform elimination for p-adic fields
zbMATH Open document ID: 0656.03023
DOI: 10.1016/0168-0072(88)90043-7
author: Luc Bélair
published in: Annals of Pure and Applied Logic
publication date: 1988-01-01T00:00:00Z
full work available at URL: APAL 1988 LB.pdf
review text: Let p be a prime. The theory \(pCF_ d\) of p-adically closed fields of p- rank d was introduced by \textit{A. Prestel} and \textit{P. Roquette} [Formally p-adic fields (1984; Zbl 0523.12016)] to study the model theory of finite extension fields (of a given degree d) of the field \({\mathbb{Q}}_ p\) of p- adic numbers. Among other things, they generalize \textit{A. Macintyre}'s [J. Symb. Logic. 41, 605-610 (1976; Zbl 0362.02046)] elimination theorem for pCF \((d=1)\) but with the addition of constants \(c_ 1,...,c_ d\) such that the \(c_ i\) yield a basis for the valuation ring modulo p as a vector space over \({\mathbb{F}}_ p\). A primitive recursive procedure for this elimination has been given by \textit{V. Weispfenning} [Lect. Notes Math. 1103, 419-472 (1984; Zbl 0584.03022)] and by \textit{J. Denef} [J. Reine Angew. Math. 369, 154-166 (1986; Zbl 0584.12015)]. For a counterexample to see that Macintyre's language alone doesn't suffice in general for \(d>1\) see \textit{E. Robinson}'s paper [J. Algebra 110, 158-172 (1987; Zbl 0623.12009)]. The results of this paper consist of an explicit axiomatization for the universal part of \(pCF_ d\) in the Macintyre-Prestel-Roquette language and a model-theoretic proof of an elimination theorem for \(Th(\{{\mathbb{Q}}_ p:\) p prime\(\})\).
Mathematics Subject Classification ID: 03C60
Mathematics Subject Classification ID: 12L12
Mathematics Subject Classification ID: 03C10
Mathematics Subject Classification ID: 11S99
zbMATH DE Number: 4070875
zbMATH Keywords: model theory of finite extension fields of the field of p-adic numbers
zbMATH Keywords: p-adically closed fields
zbMATH Keywords: axiomatization
MaRDI profile type: Publication
OpenAlex ID: W2090495484
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description: scientific article; zbMATH DE number 4070875