A refinement of the Dress–Scharlau theorem

In 1982, Dress and Scharlau [1] found an upper bound for the norm of totally positive, additively indecomposable algebraic integers in real quadratic fields and showed that this bound is sharp if the norm of the fundamental unit is −1. In this paper, we prove that their upper bound is not extremal if the norm of the fundamental unit is +1 and establish a new upper bound that is sharp for a large family of such fields.