Maximum norm of indecomposable totally positive algebraic integers

Determine the maximum norm of an indecomposable totally positive algebraic integer in a given totally real number field.

Background

For a totally real number field K, the totally positive algebraic integers form an additive semigroup whose indecomposable elements are those that cannot be expressed as sums of two totally positive algebraic integers. The paper explains that Narkiewicz formulated the problem of determining the largest norm attained by such an indecomposable element in a given field. Existing work provides upper bounds, including the discriminant bound, and examples suggest that this bound may be close to optimal, but the exact maximum is not determined in general.