Let K be a field. The étale open topology on the K-points V (K) of a K-variety V was introduced in [JTWY]. The étale open topology is non-discrete if and only if K is large. If K is separably, real, p-adically closed then the étale open topology agrees with the Zariski, order, valuation topology, respectively. We show that existentially definable sets in perfect large fields behave well with respect to this topology: such sets are finite unions of étale open subsets of Zariski closed sets. This… Expand

For an arbitrary field $K$ and $K$-variety $V$, we introduce the etale-open topology on the set $V(K)$ of $K$-points of $V$. This topology agrees with the Zariski topology, Euclidean topology,… Expand

This paper shall outline a new treatment of the p -adic case (not far removed from the classical treatments cited above), and establish an new analogy between real closed and p -adically closed fields.Expand

In this paper we study Galois theoretic properties of a large class of fields, a class which includes all fields satisfying a universal local-global principle for the existence of rational points on… Expand

We prove that the theory of a Henselian valued field of characteristic zero, with finite ramification, and whose value group is a $Z$-group, is model-complete in the language of rings if the theory… Expand

In this survey note we present the state of the art on the theory of Banach and Hilbert spaces over complete valued scalar fields that are not isomorphic to R or C. For convenience we treat… Expand

§1.1. Motivation. The purpose of these notes is to explain the definition and basic properties of the Néron model A of an abelian variety A over a global or local field K. We also give some idea of… Expand

Introduction, ultraproducts, definitions and examples elementary equivalence - axiomatizable and finitely axiomatizable classes - examples and results in field theory elementary definability -… Expand