A version of p-adic minimality

We introduce a very weak language Lp on Qp, which is just rich enough to have the same definable subsets of the line Qp than one has using the ring language. We prove that the only definable functions in the language Lp are trivial functions. We further give a definitional expansion Lp of Lp in which Qp has quantifier elimination and we obtain a weak cell decomposition. Our language Lp can serve as the p-adic analogue of the very weak language (<) on the real numbers to define a notion of minimality on the p-adics. Finally we give a universal-existential definition, in the ring language, of Zp inside Qp and of Fp[[t]] inside Fp((t)), which works uniformly in all p, and in all finite field extensions.