Euclidean ideal classes in Galois number fields of odd prime degree

Abstract

Weinberger in 1972, proved that the ring of integers of a number field with unit rank at least $1$ is a principal ideal domain if and only if it is a Euclidean domain, provided the generalised Riemann hypothesis holds. Lenstra extended the notion of Euclidean domains in order to capture Dedekind domains with finite cyclic class group and proved an analogous theorem in this setup. More precisely, he showed that the class group of the ring of integers of a number field with unit rank at least $1$ is cyclic if and only if it has a Euclidean ideal class, provided the generalised Riemann hypothesis holds. The aim of this paper is to show the following. Suppose that $\mathbf{K}_1$ and $\mathbf{K}_2$ are two Galois number fields of odd prime degree with cyclic class groups and Hilbert class fields that are abelian over $\mathbb{Q}$. If $\mathbf{K}_1\mathbf{K}_2$ is ramified over $\mathbf{K}_i$, then at least one $\mathbf{K}_i$ ($i \in \{1,2\}$) must have a Euclidean ideal class.