Year: 2026
Type: preprint
Abstract: Motivated by Clark's work on Euclidean quadratic forms, we introduce trace Euclidean and house Euclidean quadratic lattices over totally real number fields, and relate these concepts through the $p$-norm Euclidean property. We prove that, as $F$ varies over all totally real number fields of arbitrary degree, there are only finitely many positive definite trace Euclidean $\mathcal{O}_{F}$-lattices of arbitrary rank $n\ge1$, up to isometry. For every $p\in[1,\infty]$, the same finiteness holds for positive definite $p$-norm Euclidean lattices. We also classify all positive definite trace Euclidean lattices of rank one over real quadratic fields: there are exactly six isometry classes, and every such lattice is free. As a corollary, the trace Euclidean real quadratic fields are precisely $\mathbb{Q}(\sqrt{2})$, $\mathbb{Q}(\sqrt{5})$, and $\mathbb{Q}(\sqrt{13})$.
Source: arXiv (Cornell University)
Author: He, Zilong
Institution: Dongguan University of Technology
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7223952990