Vol. 15, No. 10, 2021
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The unit equation over cyclic number fields of prime degree

Nuno Freitas, Alain Kraus and Samir Siksek
Vol. 15 (2021), No. 10, 2647–2653
DOI: 10.2140/ant.2021.15.2647
Abstract

Let ℓ≠3 be a prime. We show that there are only finitely many cyclic number fields F of degree ℓ for which the unit equation

λ + μ = 1,λ,μ ∈𝒪F×

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 ℚ(ζ11)+.

Keywords
unit equation, cyclic fields, exceptional fields
Mathematical Subject Classification
Primary: 11R20
Milestones
Received: 11 December 2020
Revised: 13 March 2021
Accepted: 15 April 2021
Published: 8 February 2022
Authors
Nuno Freitas
Department de Matemàtiques i Informàtica
Universitat de Barcelona
Barcelona
Spain
Alain Kraus
Institut de Mathématiques de Jussieu - Paris Rive Gauche
Sorbonne Université
UMR 7586 CNRS - Paris Diderot
Paris
France
Samir Siksek
Mathematics Institute
University of Warwick
Coventry
United Kingdom