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

Let 3 be a prime. We show that there are only finitely many cyclic number fields F of degree 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 (ζ11)+.

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