Representaion of algebraic integers as sum of units over the real quadratic fields

Abstract EN

In this paper we generalize Jacobsons results by proving that any integer α in Q(√d),(d>0,d is a square-free integer), belong toW_t.

All units of Q(√d) are generated by the fundamental unit ε^n,(n≥0) having the forms ε=t+√d,d≢1(mod4) ε=[(2t-1)+√d]/2,d≡1(mod4) Our generalization build 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 W_1,W_2,W_3… Jacobsons results shows that Q√2,Q√5∈W_1 and Sliwa confirm that Q√2 and Q√5 are the only real quadratic fields in W_1.