Sums of two units in number fields

2025/02/03 by Tinková, Magdaléna, Visser, Robin, Yatsyna, Pavlo
#11D61 (Primary) 11J87 #11R16 #11R27 (Secondary) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2502.01345

Abstract

Let K be a number field with ring of integers O_K. Let N_K be the set of positive integers n such that there exist units ε, δ∈ O_K^× satisfying ε + δ= n. We show that N_K is a finite set if K does not contain any real quadratic subfield. In the case where K is a cubic field, we also explicitly classify all solutions to the unit equation ε + δ= n when K is either cyclic or has negative discriminant.

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