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A version of p-adic minimality

Published online by Cambridge University Press:  12 March 2014

Raf Cluckers  and
Eva Leenknegt
Raf Cluckers
Affiliation:
Université Lille 1, Laboratoiré Painlevé, CNRS - UMR 8524, Cité Scientifique, 59655 Villeneuve d'Ascq Cedex, France Katholieke Universiteit Leuven, Department of Mathematics, Celestijnenlaan 200B, B-3001 Leuven, Belgium, E-mail: Raf.Cluckers@math.univ-lille1.fr, URL: http://math.univ-lille1.fr/~cluckers
Eva Leenknegt
Affiliation:
Purdue University, Department of Mathematics, 150 N. University Street, West Lafayette, IN 47907-2067, USA, E-mail: eleenkne@math.purdue.edu, URL: http://www.math.purdue.edu/~eleenkne
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Abstract

We introduce a very weak language on p-adic fields K, which is just rich enough to have exactly the same definable subsets of the line K that one has using the ring language. (In our context, definable always means definable with parameters.) We prove that the only definable functions in the language are trivial functions. We also give a definitional expansion of in which K has quantifier elimination, and we obtain a cell decomposition result for -definable sets.

Our language can serve as a p-adic analogue of the very weak language (<) on the real numbers, to define a notion of minimality on the field of p-adic numbers and on related valued fields. These fields are not necessarily Henselian and may have positive characteristic.

Keywords

p-adic numbersquantifier eliminationcell decompositionP-minimalityo-minimalitydefinability
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 77 , Issue 2 , June 2012 , pp. 621 - 630
Copyright
Copyright © Association for Symbolic Logic 2012

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References

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