On Limits of Sequences of Algebraic Elements over a Complete Field

• 来源期刊: 代数集刊（英文版）
• 关键词: valued fields non-Archimedean valued fields completions
• 年，卷（期）: 2005,（4）
• 页码范围: 617-628
• 页数: 12页
• 语种: 英文

摘要

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