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

Wen Yao Chang, Chih Ren Cheng, Ming Guang Leu

Research output: Contribution to journalArticlepeer-review

Abstract

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.

Original languageEnglish
Pages (from-to)605-616
Number of pages12
JournalIsrael Journal of Mathematics
Volume216
Issue number2
DOIs
StatePublished - 1 Oct 2016

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