Norm-Euclidean domains

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A norm-Euclidean domain is an Euclidean domain with respect to the norm function or the absolute value of the norm function. For real quadratic integer rings (as opposed to imaginary quadratic integer rings), the absolute value of the norm must be used because the norm may be negative.

The following quadratic rings are norm-Euclidean: ${{𝒪}}_{\mathbb{\mathbb{Q}}(\sqrt{-11})}$, ${{𝒪}}_{\mathbb{\mathbb{Q}}(\sqrt{-7})}$, $\mathbb{\mathbb{Z}}[\omega ]$, $\mathbb{\mathbb{Z}}[\sqrt{-2}]$, $\mathbb{\mathbb{Z}}[i]$, $\mathbb{\mathbb{Z}}[\sqrt{2}]$, $\mathbb{\mathbb{Z}}[\sqrt{3}]$, $\mathbb{\mathbb{Z}}[\varphi ]$, $\mathbb{\mathbb{Z}}[\sqrt{6}]$, $\mathbb{\mathbb{Z}}[\sqrt{7}]$, $\mathbb{\mathbb{Z}}[\sqrt{11}]$, ${{𝒪}}_{\mathbb{\mathbb{Q}}(\sqrt{13})}$, ${{𝒪}}_{\mathbb{\mathbb{Q}}(\sqrt{17})}$, $\mathbb{\mathbb{Z}}[\sqrt{19}]$, ${{𝒪}}_{\mathbb{\mathbb{Q}}(\sqrt{21})}$, ${{𝒪}}_{\mathbb{\mathbb{Q}}(\sqrt{29})}$, ${{𝒪}}_{\mathbb{\mathbb{Q}}(\sqrt{33})}$, ${{𝒪}}_{\mathbb{\mathbb{Q}}(\sqrt{37})}$, ${{𝒪}}_{\mathbb{\mathbb{Q}}(\sqrt{41})}$, ${{𝒪}}_{\mathbb{\mathbb{Q}}(\sqrt{57})}$, ${{𝒪}}_{\mathbb{\mathbb{Q}}(\sqrt{73})}$ (see A048981).

References

• Ian Stewart & David Tall, Algebraic Number Theory and Fermat's Last Theorem, 3rd Ed. Natick, Massachusetts: A. K. Peters (2002): 99