We introduce a very weak language ℒM 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 ℒM are trivial functions. We also give a definitional expansion ℒM' of ℒM in which K has quantifier elimination, and we obtain a cell decomposition result for ℒM-definable sets.
Our language ℒM 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.
"A version of p-adic minimality." J. Symbolic Logic 77 (2) 621 - 630, June 2012. https://doi.org/10.2178/jsl/1333566641