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

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.

💡 Research Summary

The paper presents explicit Euclidean division algorithms for the six norm‑Euclidean real quadratic fields ℚ(√m) with m ∈ {2, 3, 6, 7, 11, 19}. For each field the authors construct an “M₁‑division algorithm”, i.e. a procedure that, given any element ξ∈K, finds an integer γ∈O_K such that |Norm(ξ−γ)|≤M₁(K), where M₁(K) is the first Euclidean minimum of the field. The key idea is to work in the fundamental square S₀=