Maximal \(RC_ \pi\)-fields (Q1346912)

The paper is a continuation of the research initiated by the author [Algebra Logic 31, 342-360 (1992); translation from Algebra Logika 31, 592-623 (1992; Zbl 0791.12005)]. Let \(\langle F,\pi \rangle\) be a formally \(\pi\)-adic field. An algebraic extension \(F_ 0\geq F\) is called a totally \(\pi\)-adic extension of \(\langle F,\pi \rangle\) if for any \(R\in W_ \pi (F)\) the field \(F_ 0\) is \(F\)-embeddable into \(H_ R (F)\). The field \(\langle F,\pi \rangle\in RC_ \pi\) is called a maximal \(RC_ \pi\)-field (\(m RC_ \pi\)-field) if \(F\) has no proper totally \(\pi\)-adic extensions. It is shown that the class of \(mRC_ \pi\)-fields is axiomatizable in the signature \(\sigma_ R\cup \langle \pi \rangle\). Denote by \({\mathfrak F}_ *\) the class of all fields \(F\) of characteristic 0 such that \(F\in m RC_ Z\), and for \(\pi\in F\) satisfying \(F\models \varphi_ 2 (\pi)\), \(\overline {R} \rightleftharpoons R_ \pi/ \pi R_ \pi\in {\mathcal E}\) and \(\pi p^{-1}\in R_ \pi\) for all prime \(p\). Here \({\mathcal E}\) is the class of elementary regular rings, specified and treated in [the author, Algebra Logic 32, No. 4, 206-214 (1993); translation from Algebra Logika 32, No. 4, 387-401 (1993)]. Various sufficient conditions are established for two fields \(F_ 1, F_ 2\in {\mathfrak F}_ *\) to be elementarily equivalent. Basic model-theoretic properties of fields in the class \({\mathfrak F}_ *\) are established. These results are used to obtain new classes of fields with decidable theories.