Abstract
In this paper we generalize Jacobson's results by proving that any integer∝in Q(√d), (d > 0,d is a square-free integer), belong to ����. All units of Q(√d) are generated by the fundamental unit ��ε^n, (n ≥ 0) having the forms: ε= t+ √d,d ≢ 1 (mod 4) ε= [(2��−1) + √d]², �� ≡ 1 (mod 4) Our generalization builds on using the conditions: t+ 1 =ε ±ε⁻¹ + (1 −t), t=ε ± ε⁻¹ + (1 − t). This leads us to classify the real quadratic fields Q(√d) into the sets W₁, W₂,W₃... Jacobson's results show that Q(√2),Q(√5)ϵ W₁ and Sliwa confirms that Q(√2) and Q(√5) are the only real quadratic fields in W₁.
Keywords
Fundamental units of real quadratic field, Integers of real quadratic field as sum of finite units, Real quadratic fields.
Article Type
Supplemental Issue
How to Cite this Article
Baddai, Saad A.
(2019)
"Representation of Algebraic Integers as Sum of Units over the Real Quadratic Fields,"
Baghdad Science Journal: Vol. 16:
Iss.
3, Article 33.
DOI: https://doi.org/10.21123/bsj.2019.16.3(Suppl.).0781
