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arXiv · 2012.06445

On the unit equation over cyclic number fields of prime degree

Nuno Freitas·Alain Kraus·Samir Siksek
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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 $\mathbb{Q}(\zeta_{11})^+$.

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Nuno Freitas, Alain Kraus, Samir Siksek. 2020-12-11. On the unit equation over cyclic number fields of prime degree. https://doi.org/10.2140/ant.2021.15.2647

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