The geometric theory of \(p\)-adic fields
type: Item
instance of: scholarly article
title: The geometric theory of \(p\)-adic fields
zbMATH Open document ID: 0623.12009
DOI: 10.1016/0021-8693(87)90041-X
published in: Journal of Algebra
publication date: 1987-01-01T00:00:00Z
review text: The paper is devoted to the extension of the well known analogy between real closed and p-adically closed fields. It gives a \(p\)-adic analogue to the finiteness theorem of semi-algebraic geometry, through a syntactic characterization of the definable closed sets (theorem 1.2). It contains an axiomatization of the subfields of a \(p\)-adically closed field, given in a language for which there is quantifier elimination, called \(p\)-adic fields. This axiomatization implies a change in the philosophy of analogy between the real and the \(p\)-adic case: the important notion is to study the axiomatization of sets of \(n\)-th powers, as natural \(p\)-adic analogues to the half line, and the valuation plays a quite incidential role. This leads to a situation more closely analogous to the theory of ordered fields: the theory of \(p\)-adically closed fields is the model completion of the theory of \(p\)-adic fields and we have the following result, similar to the relation between real and real closed fields: given a \(p\)-adic field \(K\) and a choice of its subsets of \(n\)-th powers \(R_n\), there exists a unique \(p\)-adic closure of \(K^*\) such that the trace on \(K\) of the subsets of \(n\)-th powers of \(K^* R^*_n\) coincides with \(R_n\).
Mathematics Subject Classification ID: 11S99
Mathematics Subject Classification ID: 12L12
Mathematics Subject Classification ID: 14G20
Mathematics Subject Classification ID: 03C10
Mathematics Subject Classification ID: 12J25
Mathematics Subject Classification ID: 12J20
zbMATH DE Number: 4010637
zbMATH Keywords: p-adically closed fields
zbMATH Keywords: definable closed sets
zbMATH Keywords: quantifier elimination
zbMATH Keywords: p-adic fields
zbMATH Keywords: sets of n-th powers
zbMATH Keywords: model completion
author: Edmund P. Robinson
reviewed by: Marie-Françoise Roy
MaRDI profile type: Publication
cites work: On definable subsets of p-adic fields
cites work: Formally \(p\)-adic fields
cites work: Q3324077
full work available at URL: 0021-8693(87)90041-x
OpenAlex ID: W2025242054
description: scientific article; zbMATH DE number 4010637