Bélair, L.; van den Dries, Lou; Macintyre, A. Elementary equivalence and codimension in p-adic fields. (English) Zbl 0665.12027 Manuscr. Math. 62, No. 2, 219-225 (1988). 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 Keywords:model theory of local fields PDFBibTeX XMLCite \textit{L. Bélair} et al., Manuscr. Math. 62, No. 2, 219--225 (1988; Zbl 0665.12027) Full Text: DOI EuDML 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 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.