Representation 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 ? 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 ?1.