Quantifier elimination in tame infinite \(p\)-adic fields (Q2758074)

Given an infinite algebraic extension \(K\) of \({\mathbb Q}_p\) which is tame, i.e., \(K\) has only tamely ramified algebraic extensions, such that the residue field \(\overline K\) satisfies Kaplansky's condition, i.e., \(\overline K\) does not admit finite extensions of \(p\)-divisible degree, the author proves that the theory of \(K\) admits quantifier elimination in the language of valued fields enlarged by the power predicates \(P_n\) introduced by \textit{A. J. Macintyre} [J. Symb. Log. 41, 605-610 (1976; Zbl 0362.02046)], and further specific predicates and constants for \(\overline K\). In particular, if \(\overline K\) is algebraically closed, then an enlargement by the predicates \(P_n\) is sufficient. Notice that a more general result on the relative elimination of quantifiers for Henselian fields of characteristic \(0\) was obtained earlier by the reviewer [Ann. Pure Appl. Log. 53, 51-74 (1991; Zbl 0734.03021)], extended later by \textit{F.-V. Kuhlmann} [Isr. J. Math. 85, 277-306 (1994; Zbl 0809.03028)]. It could be useful to try to recover the main result of the paper under review from the more general one mentioned above.