On the closed subfields of \(C_p\)
Let \(p\) be a prime integer and \(\mathbb{Q}_p\) be the field of \(p\)-adic numbers and \(v\) the valuation of \(\mathbb{Q}_p\) characterised by \(v(p) =1\). Denote by \(\mathbb{Q}_p\) a fixed algebraic closure of \(\mathbb{Q}_p\) and also by \(v\) the unique extension of \(v\) to \(\overline {\mathbb{Q}}_p\). Let \((C_p,v)\) denote the completion of \((\mathbb{Q}_p,v)\). In an earlier paper \textit{A. Iovita} and \textit{A. Zaharescu} [cf. J. Number Theory 50, 202-205 (1995; Zbl 0813.12006)] proved that if \(K\) is a closed subfield of \(\mathbb{C}_p\) containing \(\mathbb{Q}_p\) which is not an algebraic extension of \(\mathbb{Q}_p\), then there exists \(t\) in \(K\) such that \(K\) is the completion of \(\mathbb{Q}_p (t)\). In this paper they show that \(t\) can be obtained as the limit of a certain well defined sequence. They introduce a particular kind of generic transcendental elements and give a criterion for two such elements to be conjugate. They have also shown that for \(t\in C_p\), \(\mathbb{Q}_p (t)\) and \(\mathbb{Q}_p [t]\) have the same topological closure.
scientific article; zbMATH DE number 1121272