All totally real abelian number fields of class number one are Euclidean

It is a famous result of Narkiewicz that there are at most two real quadratic and at most one totally real cubic Galois extensions $K\mathrm{/}\mathbb{Q}$, whose ring of integers ${\mathcal{O}}_K$ is a unique factorization domain (UFD), but not Euclidean. As predicted by another famous result due to Weinberger, these exceptions should in fact not exist, as their existence would contradict a generalized version of the Riemann hypothesis. Building on recent spectacular progress on the Quasi-Riemann hypothesis for Hecke characters, in this short note we close the above final gap provided by Narkiewicz’s result, and show, more generally, that the ring of integers ${\mathcal{O}}_K$ of any totally real abelian number field $K$ is a UFD, if and only if it is Euclidean.