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

$$\lambda +\mu =1,\lambda ,\mu \in {\mathcal{𝒪}}_F^{\times }$$

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 $\mathbb{Q}{\left({\zeta }_{11}\right)}^{+}$.

中文翻译：

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素数循环数域上的单位方程

让$\ell \ne 3$成为素数。我们证明只有有限多个循环数域$F$学位$\ell$其中单位方程

$$\lambda +\mu =1,\lambda ,\mu \in {\mathcal{𝒪}}_F^{\times }$$

有解决方案。我们的结果是有效的。例如，我们推论出单位方程有解的唯一循环五次数域是$\mathbb{Q}{\left({\zeta }_{11}\right)}^{+}$.