Trace and norm of indecomposable integers in cubic orders

Additively indecomposable algebraic integers are a useful tool in the study of universal quadratic forms over totally real number fields. Except for some results for real quadratic, real biquadratic, and some totally real cubic fields, we know almost nothing about these elements. In this talk, we will focus on the upper bound on their norm and, moreover, on their traces when we multiply them by elements of the codifferent.