Division algorithms for norm-Euclidean real quadratic fields -- part I

TL;DR

The paper tackles explicit $M_1$-division algorithms for norm-Euclidean real quadratic fields $\mathbb{Q}(\sqrt{m})$ with $m\in\{2,3,6,7,11,19\}$ by constructing geometric coverings of the centered square via hyperbolas defined by the norm $f_m(a,b)=a^2-mb^2$ and proving coverage with exact computations. The authors develop a toolkit for exact arithmetic (rounding, sign of quadratics, sign of sums of radicals) and formulate an informal algorithm that iteratively covers $\mathcal{S}_0=[0,1/2]^2$ with regions $\mathcal{H}_{u,v}$, yielding explicit $\gamma=x+y\sqrt{m}$ satisfying $|\mathrm{Norm}(\xi-\gamma)|\le M$ for $\xi=a+b\sqrt{m}$. They provide detailed, case-by-case constructions for $m=2,3,6,7,11,19$, including the particularly intricate $m=19$ running through numerous steps and data files to ensure reproducibility; these constitute concrete $M_1$-division algorithms, not merely existence proofs. The results enable fast and reliable gcd-type computations in these fields and lay groundwork for extensions to $m\equiv1\pmod{4}$ and to higher-degree settings, with an emphasis on rigorous, exact verification. Overall, the work advances practical explicit division algorithms in real quadratic fields through geometry-guided, exact computations and a reproducible computational framework.

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.

Figures (16)

• Figure 1: Sign of a quadratic polynomial on $[x_{\min}, x_{\max}]$.
• Figure 2: The cases $m=2$ (left), $m=3$ (right).
• Figure 3: The case $m=6$ (right).
• Figure 4: The case $m=7$: step 1 (left) and step 2 (right).
• Figure 5: The case $m=7$: step 3.

Theorems & Definitions (14)

• Proposition 1.1
• Lemma 2.1
• Lemma 2.2
• Proposition 2.3
• Remark 2.1
• Remark 2.2
• Proposition 3.1
• Definition 3.2
• Lemma 3.3
• Proposition 3.4