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Weinberger in 1972, proved that the ring of integers of a number field withunit rank at least $1$ is a principal ideal domain if and only if it is aEuclidean domain, provided the generalised Riemann hypothesis holds. Lenstraextended the notion of Euclidean domains in order to capture Dedekind domainswith 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 anumber field with unit rank at least $1$ is cyclic if and only if it has aEuclidean ideal class, provided the generalised Riemann hypothesis holds. Theaim 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 cyclicclass 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.

Euclidean ideal classes in Galois number fields of odd prime degree