Low degree extensions with Cyclic class group
Srilakshmi Krishnamoorthy, Sunil Kumar Pasupulati
TL;DR
The paper extends Lenstra's framework of Euclidean ideal classes to abelian real quartic and biquadratic fields, establishing existence results under explicit Galois-conductor conditions and leveraging density arguments for primes with structured factorizations. It provides concrete corollaries for biquadratic fields, easing prior restrictions such as congruence requirements, and extends the analysis to real cubic and quadratic fields by genus-theoretic reasoning, often removing the need for prime-class-number hypotheses. A conditional result under the Elliott–Halberstam conjecture further strengthens the occurrence of Euclidean ideal classes in abelian real cubic fields, using prime progressions and primitive root considerations. The work offers a catalog of explicit examples and computational confirmations, highlighting new instances where Euclidean ideals exist and advancing understanding of when rings of integers in number fields admit Euclidean reductions without GRH.
Abstract
Lenstra introduced the notion of the Euclidean ideal class, a generalization of the Euclidean domain that captures cyclic class groups. In this article, we establish the existence of Euclidean ideal classes in abelian quartic fields. As a corollary, we demonstrate that certain biquadratic fields with class number two possess a Euclidean ideal class. Additionally, we investigate the presence of Euclidean ideal classes in specific cubic and quadratic extensions.