PreprintCC BY 4.0arXiv
Orders in the Euclidean Court
Abstract
Following Johnson, Queen, and Sevilla, we call an order 𝒪 in a number field K \textsf{generalized Euclidean} if for all α,β∈ 𝒪 with β≠ 0 for which α𝒪+β𝒪 is a principal ideal of 𝒪, there are γ, ρ∈ 𝒪 with α=βγ+ ρ and |NK/ℚ(ρ)| < |NK/ℚ(β)|. We prove that only finitely many quadratic orders are generalized Euclidean. Additionally, assuming GRH, we show that for each fixed d > 1 there are only finitely many integers m for which Td−m is irreducible over ℚ and ℤ[m1/d] = ℤ[T]/(Td−m) is generalized Euclidean. When d is even, we can remove the assumption of GRH.
Related
BibTeX
@misc{adajar2026orders,
title = {{Orders in the Euclidean Court}},
author = {Carlo Francisco Adajar and Kevin J. McGown and Paul Pollack and Enrique Treviño},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.25793},
eprint = {2610.25793},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.25793},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Orders in the Euclidean Court
AU - Carlo Francisco Adajar
AU - Kevin J. McGown
AU - Paul Pollack
AU - Enrique Treviño
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.25793
UR - https://arxiv.org/abs/2610.25793
AB - Following Johnson, Queen, and Sevilla, we call an order 𝒪 in a number field K \textsf{generalized Euclidean} if for all α,β∈ 𝒪 with β≠ 0 for which α𝒪+β𝒪 is a principal ideal of 𝒪, there are γ, ρ∈ 𝒪 with α=βγ+ ρ and |N_K/ℚ(ρ)| < |N_K/ℚ(β)|. We prove that only finitely many quadratic orders are generalized Euclidean. Additionally, assuming GRH, we show that for each fixed d > 1 there are only finitely many integers m for which T^d−m is irreducible over ℚ and ℤ[m^1/d] = ℤ[T]/(T^d−m) is generalized Euclidean. When d is even, we can remove the assumption of GRH.
ER -