On distinct unit generated fields that are totally complex

TL;DR

This paper characterizes totally complex number fields where every algebraic integer can be expressed as a sum of distinct units, extending previous methods to include totally complex quartic fields.

Contribution

It extends a method to characterize such fields, specifically applying it to totally complex quartic number fields for the first time.

Findings

Identified conditions for totally complex fields where algebraic integers are sums of distinct units.

Extended existing methods to handle totally complex quartic fields.

Provided new characterizations for these number fields.

Abstract

We consider the problem of characterizing all number fields $K$ such that all algebraic integers $\alpha\in K$ can be written as the sum of distinct units of $K$. We extend a method due to Thuswaldner and Ziegler that previously did not work for totally complex fields and apply our results to the case of totally complex quartic number fields.