#### 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 $\ell \ne 3$ be a prime. We show that there are only finitely many cyclic number fields $F$ of degree $\ell$ for which the unit equation

$\lambda +\mu =1,\phantom{\rule{0.3em}{0ex}}\phantom{\rule{1em}{0ex}}\lambda ,\mu \in {\mathsc{𝒪}}_{F}^{×}$

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 $ℚ{\left({\zeta }_{11}\right)}^{+}$.

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