TY - JOUR
T1 - A remark on the ring of algebraic integers in Q(√-d)
AU - Chang, Wen Yao
AU - Cheng, Chih Ren
AU - Leu, Ming Guang
N1 - Publisher Copyright:
© 2016, Hebrew University of Jerusalem.
PY - 2016/10/1
Y1 - 2016/10/1
N2 - 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.
AB - 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.
UR - http://www.scopus.com/inward/record.url?scp=84991489195&partnerID=8YFLogxK
U2 - 10.1007/s11856-016-1421-7
DO - 10.1007/s11856-016-1421-7
M3 - 期刊論文
AN - SCOPUS:84991489195
SN - 0021-2172
VL - 216
SP - 605
EP - 616
JO - Israel Journal of Mathematics
JF - Israel Journal of Mathematics
IS - 2
ER -