Division algorithms for norm-Euclidean real quadratic fields -- part I
Abstract
We give a Euclidean division algorithm for the real quadratic fields $\mathbb{Q}(\sqrt{m})$ for $m \in \{2, 3, 6, 7, 11, 19\}$, with the property that the norm of the remainder depends on the first Euclidean minimum of the field. In each case, we cover the square $[-1/2, 1/2] \times [-1/2, 1/2]$ with hyperbolas and give a list of these, together with regions covered. We mechanize the proofs as much as we can, using exact computations, in order to be able to reproduce them.
Fields of Study & Subjects
Algebraic Geometry and Number Theory
Analytic Number Theory Research
Advanced Algebra and Geometry
Publication details
- Journal / source
- arXiv (Cornell University)
- Publication date
- 6 February 2026
- Language
- EN
- Document type
- Preprint