  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.
