On the model companion for \(e\)-fold \(p\)-adically valued fields (Q1191473)

The paper under review corresponds to a chapter in the author's Ph. D. thesis (Tel-Aviv Univ.). Let \({\mathcal L}_ e\), \(e\geq 0\), be the first- order language of rings augmented by \(e\) unary predicate symbols \({\mathcal O}_ 1,\dots,{\mathcal O}_ e\). The author considers the \({\mathcal L}_ e\)- theory \(T_ e\) whose models are the structures \((E,O_ 1,\dots,O_ e)\) in which \(E\) is a field and \(O_ 1,\dots,O_ e\) are \(p\)-adic valuation rings on \(E\). Let \(\overline {T}_ e\) be a model-companion for \(T_ e\). \(\overline{T}_ e\) is a model-complete \({\mathcal L}_ e\)-theory such that each model of \(\overline{T}_ e\) is a model of \(T_ e\) and each model of \(T_ e\) extends to a model of \(\overline{T}_ e\). A criterion for elementary equivalence of \(p\)-adically maximal pseudo \(p\)-adically closed fields is proved. The criterion is applied to show the existence of the model-companion of the theory \(T_ e\) and to describe its models. The author obtains three characterizations of \(\overline{T}_ e\) by means of geometro-algebraic, Galois-theoretic and measure-theoretic properties. Earlier van den Dries has proved the existence of a model-companion for the theory of \(e\)-fold \(p\)-adically valued fields in an extended language containing Macintyre predicates [\textit{L. P. D. van den Dries}, Model theory of fields (Thesis, Utrecht 1978)].