The geometric theory of \(p\)-adic fields
The paper is devoted to the extension of the well known analogy between real closed and p-adically closed fields. It gives a \(p\)-adic analogue to the finiteness theorem of semi-algebraic geometry, through a syntactic characterization of the definable closed sets (theorem 1.2). It contains an axiomatization of the subfields of a \(p\)-adically closed field, given in a language for which there is quantifier elimination, called \(p\)-adic fields. This axiomatization implies a change in the philosophy of analogy between the real and the \(p\)-adic case: the important notion is to study the axiomatization of sets of \(n\)-th powers, as natural \(p\)-adic analogues to the half line, and the valuation plays a quite incidential role. This leads to a situation more closely analogous to the theory of ordered fields: the theory of \(p\)-adically closed fields is the model completion of the theory of \(p\)-adic fields and we have the following result, similar to the relation between real and real closed fields: given a \(p\)-adic field \(K\) and a choice of its subsets of \(n\)-th powers \(R_n\), there exists a unique \(p\)-adic closure of \(K^*\) such that the trace on \(K\) of the subsets of \(n\)-th powers of \(K^* R^*_n\) coincides with \(R_n\).
scientific article; zbMATH DE number 4010637