A version of p-adic minimality

Abstract

We introduce a very weak language ℒ_M 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 ℒ_M are trivial functions. We also give a definitional expansion ℒ_M' of ℒ_M in which K has quantifier elimination, and we obtain a cell decomposition result for ℒ_M-definable sets.

Our language ℒ_M 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.