Abstract
In this paper we generalize Jacobsons results by proving that any integer in is a square-free integer), belong to . All units of are generated by the fundamental unit having the forms
 Our generalization build on using the conditions
 This leads us to classify the real quadratic fields into the sets Jacobsons results shows that and Sliwa confirm that and are the only real quadratic fields in .
Highlights
Thereal quadratic fields Q(θ) of degree two over the rational numbers with θ = d, where d > 0 and d is a square free integer for if θ = √d is a root of the quadratic polynomial x2 + 2ax + b = 0 This leads to state that any integer α ∈ Q(√d) has the forms α = a + b√d, d ≢ 1(mod4) or α = (a + b√d)/2, d ≡ 1(mod4) The fundamental unit of Q(√d) has the forms ε = t + √d, d ≢ 1(mod4)
Are written as a sum of distinct units, where ε1 = 1 + √2, ε2 = (1 + √5)/2 are the fundamental units of Q(√2) and Q(√5)
Def (2.1): We define Wt, t ≥ 1 to be the set of all real quadratic fields Q(√d) in which every integer α ∈ Q(√d) is represented as a sum of units with at most t- repititions
Summary
Representaion of Algebraic Integers as Sum of Units over the Real Quadratic Fields Def (2.1): We define Wt, t ≥ 1 to be the set of all real quadratic fields Q(√d) in which every integer α ∈ Q(√d) is represented as a sum of units with at most t- repititions.
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