Let us call an integer part of an ordered field any subring such that every element of the field lies at distance less than 1 from a unique element of the ring. We show that every real closed field has an integer part.

We show basic facts about dp-minimal ordered structures. The main results are: dp-minimal groups are abelian-by-finite-exponent, in a divisible ordered dp-minimal group, any infinite set has non-empty interior, and any theory of pure tree is dp-minimal.

We study first-order properties of the quotient rings C(V)/P by a prime ideal P, where C(V) is the ring of p-adic valued continuous definable functions on some affine p-adic variety V. We show that they are integrally closed Henselian local rings, with a p-adically closed residue field and field of fractions, and they are not valuation rings in general but always satisfy ∀ x, y(x|y 2 ∨ y|x 2 ).

C-minimality is a variant of o-minimality in which structures carry, instead of a linear ordering, a ternary relation interpretable in a natural way on set of maximal chains of a tree. This notion is discussed, a cell-decomposition theorem for C-minimal structures is proved, and a notion of dimension is introduced. It is shown that C-minimal fields are precisely valued algebraically closed fields. It is also shown that, if certain specific ‘bad’ functions are not definable, then algebraic closure has the exchange (...) property, and for definable sets dimension coincides with the rank obtained from algebraic closure. (shrink)