Separably closed valued fields: immediate expansions
It is a key model theoretic question to delimit the expressive definability power of various first-order structures on a set. Motivated by results of \textit{D. Haskell} and \textit{D. Macpherson} [J. Symb. Log. 63, No. 2, 739--741 (1998; Zbl 0920.03047)], the author introduces the natural notion of ``immediate expansion'' of first-order structures. The motivating examples are algebraically closed valued fields and real closed convexly valued fields: they are immediate expansions of the underlying algebraically closed fields and real closed fields, as any definable set in the valued structure is either definable in the pure field structure or the valuation ring is definable from that definable set. The main result of the paper is an analogue result for separably closed valued fields of any imperfection degree. The main tools are the author's quantifier elimination for separably closed valued fields [J. Symb. Log. 81, No. 3, 887--900 (2016; Zbl 1432.03062)] and Weil's restriction. To this reviewer's knowledge, it is unknown if a suitable non-trivial \(p\)-adic analogue holds.
scientific article; zbMATH DE number 6673872
scientific article