Substructures and uniform elimination for p-adic fields
Let p be a prime. The theory \(pCF_ d\) of p-adically closed fields of p- rank d was introduced by \textit{A. Prestel} and \textit{P. Roquette} [Formally p-adic fields (1984; Zbl 0523.12016)] to study the model theory of finite extension fields (of a given degree d) of the field \({\mathbb{Q}}_ p\) of p- adic numbers. Among other things, they generalize \textit{A. Macintyre}'s [J. Symb. Logic. 41, 605-610 (1976; Zbl 0362.02046)] elimination theorem for pCF \((d=1)\) but with the addition of constants \(c_ 1,...,c_ d\) such that the \(c_ i\) yield a basis for the valuation ring modulo p as a vector space over \({\mathbb{F}}_ p\). A primitive recursive procedure for this elimination has been given by \textit{V. Weispfenning} [Lect. Notes Math. 1103, 419-472 (1984; Zbl 0584.03022)] and by \textit{J. Denef} [J. Reine Angew. Math. 369, 154-166 (1986; Zbl 0584.12015)]. For a counterexample to see that Macintyre's language alone doesn't suffice in general for \(d>1\) see \textit{E. Robinson}'s paper [J. Algebra 110, 158-172 (1987; Zbl 0623.12009)]. The results of this paper consist of an explicit axiomatization for the universal part of \(pCF_ d\) in the Macintyre-Prestel-Roquette language and a model-theoretic proof of an elimination theorem for \(Th(\{{\mathbb{Q}}_ p:\) p prime\(\})\).
model theory of finite extension fields of the field of p-adic numbers
scientific article; zbMATH DE number 4070875
scientific article