Chevalley's class number formula, unit equations and the asymptotic Fermat's Last Theorem

Abstract

Let $F$ be a number field and ${\mathcal{O}}_F$ its ring of integers. We use Chevalley's ambiguous class number formula to give a criterion for the non-existence of solutions to the unit equation $\lambda +\mu =1$, $\lambda ,\mu \in {\mathcal{O}}_F^{\times }$. This is then used to strengthen a criterion for the asymptotic Fermat's Last Theorem due to Freitas and Siksek.