Abstract
In this paper we generalize Jacobsons results by proving that any integer �� in ��(√��),(��>0,�� is a square-free integer), belong to����. All units of ��(√��) are generated by the fundamental unit ����,(��≥0) having the forms � �=��+√��,��≢1(������4) � �=[(2��−1)+√��] 2 ,��≡1(������4) Our generalization build on using the conditions � �+1=��±��−1+(1−��), � �=��±��−1+(1−��). This leads us to classify the real quadratic fields ��√�� into the sets ��1,��2,��3… Jacobsons results shows that ��√2,��√5∈��1 and Sliwa confirm that ��√2 and ��√5 are the only real quadratic fields in W1.
Keywords
Real quadratic fields, Fundamental units of real quadratic field, Integers of real quadratic field as sum of finite units.
Article Type
Supplemental Issue
How to Cite this Article
baddai, saad A.
(2020)
"Representaion of Algebraic Integers as Sum of Units over the Real Quadratic Fields,"
Baghdad Science Journal: Vol. 17:
Iss.
1, Article 30.
DOI: https://doi.org/10.21123/bsj.2020.17.1(Suppl.).0348
