Alexandru, Victor; Popescu, Nicolae; Zaharescu, Alexandru On the closed subfields of \(C_p\). (English) Zbl 0901.11035 J. Number Theory 68, No. 2, 131-150 (1998). 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 A. Iovita and 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. Reviewer: S.K.Khanduja (Chandigarh) Cited in 3 ReviewsCited in 27 Documents MSC: 11S99 Algebraic number theory: local fields 12F05 Algebraic field extensions Keywords:closed subfields of \(C_p\); \(p\)-adic fields; algebraic extensions; generic transcendental elements; topological closure Citations:Zbl 0813.12006 × Cite Format Result Cite Review PDF Full Text: DOI References: [1] Alexandru, V.; Popescu, A.; Popescu, N., Completion of r.t. extensions of local fields (I), Math. Zeitschrift, 221, 675-682 (1996) · Zbl 0852.12003 [2] Alexandru, V.; Popescu, N.; Zaharescu, A., A theorem of characterization of residual transcendental extensions of a valuation, J. Math. 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