A remark on the ring of algebraic integers in Q(√-d)

Wen Yao Chang, Chih Ren Cheng, Ming Guang Leu

研究成果: 雜誌貢獻期刊論文同行評審

摘要

It is well-known that the rings Od of algebraic integers in (√-d) for d = 19, 43, 67, and 163 are principal ideal domains but not Euclidean. In this article we shall provide a method, based on a result of P. M. Cohn, to construct explicitly pairs (b, a) of integers in Od for d = 19, 43, 67, and 163 such that, in Od, there exists no terminating division chain of finite length starting from the pairs (b, a). That is, a greatest common divisor of the pairs (b, a) exists in Od but it can not be obtained by applying a terminating division chain of finite length starting from (b, a). Furthermore, for squarefree positive integer d ∉ {1, 2, 3, 7, 11, 19, 43, 67, 163}, we shall also construct pairs (b, a) of integers in Od which generate Od but have no terminating division chain of finite length. It is of interest to note that our construction provides a short alternative proof of a theorem of Cohn which is related to the concept of GE2-rings.

原文???core.languages.en_GB???
頁(從 - 到)605-616
頁數12
期刊Israel Journal of Mathematics
216
發行號2
DOIs
出版狀態已出版 - 1 10月 2016

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