Euclidean ideal classes in Galois number fields of odd prime degree

TL;DR

This paper investigates Euclidean ideal classes in Galois number fields of odd prime degree, establishing conditions under which such fields must possess Euclidean ideal classes based on ramification and class group properties.

Contribution

It proves that for two Galois number fields of odd prime degree with cyclic class groups, ramification implies at least one field has a Euclidean ideal class, extending previous results.

Findings

If two Galois fields of odd prime degree with cyclic class groups ramify over each other, at least one has a Euclidean ideal class.

The result relies on properties of Hilbert class fields being abelian over .

Extends classical results relating class groups and Euclidean domains to specific Galois extensions.

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.