On limits of sequences of algebraic elements over a complete field

Abstract

Let K be a complete field with respect to a real non-trivial valuation v, and ν̅be the extension of v to an algebraic closure K̅ of K. A well-known result of Ostrowski asserts that the limit of a Cauchy sequence of elements of K̅ does not always belong to K̅ unless K̅is a finite extension of K. In this paper, it is shown that when a Cauchy sequence { b_n} of elements of K̅ is such that the sequence { [K(b_n): K] } of degrees of the extensions K(b_n)/K does not tend to infinity as n approaches infinity, then {b_n}has a limit in K̅.We also give a characterization of those Cauchy sequences {b_n} of elements of K̅whose limit is not in K̅,which generalizes a result of Alexandru, Popescu and Zaharescu.