On the unit equation over cyclic number fields of prime degree

TL;DR

This paper proves finiteness results for solutions to the unit equation in cyclic number fields of prime degree, providing effective bounds and identifying specific fields where solutions occur.

Contribution

It establishes the finiteness of cyclic prime degree number fields with solutions to the unit equation and determines the unique such field for degree five.

Findings

Finiteness of cyclic prime degree fields with solutions

Effective bounds for solutions

Identification of the unique cyclic quintic field with solutions

Abstract

Let $\ell \ne 3$ be a prime. We show that there are only finitely many cyclic number fields $F$ of degree $\ell$ for which the unit equation $$\lambda + \mu = 1, \qquad \lambda,~\mu \in \mathcal{O}_F^\times$$ has solutions. Our result is effective. For example, we deduce that the only cyclic quintic number field for which the unit equation has solutions is $\mathbb{Q}(\zeta_{11})^+$.