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Elementary equivalence and codimension in p-adic fields. (English) Zbl 0665.12027

From the introduction: “It is well known that an algebraically closed field of characteristic zero contains a real closed field of codimension 2. From the point of view of the model theory of real closed fields, this means that a field elementarily equivalent to a finite extension of the real numbers \({\mathbb{R}}\) contains a subfield of the right codimension elementarily equivalent to \({\mathbb{R}}.\)
We show that this is not the case for any field K finite-dimensional over \({\mathbb{Q}}_ p\). Namely, for any finite extension of \({\mathbb{Q}}_ p\) of a given degree \(d>1\), there exists an elementarily equivalent field which does not contain a subfield of finite codimension elementarily equivalent to \({\mathbb{Q}}_ p\).... Our arguments rely on basic algebraic-geometric facts, together with the completeness of \({\mathbb{Q}}_ p\) via Baire’s theorem.”
Reviewer: W.Veldman

MSC:

12L12 Model theory of fields
03C60 Model-theoretic algebra
14G20 Local ground fields in algebraic geometry
11S85 Other nonanalytic theory
12J25 Non-Archimedean valued fields
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References:

[1] A. Prestel and P. Roquette.Formally p-adic Fields, Springer-Verlag, LNM 1050, 1984 · Zbl 0523.12016
[2] A. Weil. Adèles and Algebraic Groups. Institute for Advanced Study, Princeton, 1961 · Zbl 0109.02101
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