Examples of norm-Euclidean ideal classes (Q2909037)

In this article, various rings with norm-Euclidean ideal classes are constructed, and others are shown not to possess a norm-Euclidean ideal class. In particular, the ring of integers in \(\mathbb Q(\sqrt{2},\sqrt{35}\,)\) does not have a norm-Euclidean ideal class although \textit{H. Graves} [Int. J. Number Theory 7, No. 8, 2269--2271 (2011; Zbl 1256.11059)] has shown that it does have a Euclidean ideal class. The main result is the classification of all pure cubic number fields with a norm-Euclidean ideal class: these are the rings of integers in the fields \(\mathbb Q(\root 3 \of{n}\,)\) with \(n = 2\), \(3\) and \(10\), and in this case, the rings are norm-Euclidean by results due to \textit{V. G. Cioffari} [Math. Comput. 33, 389--398 (1979; Zbl 0399.12001)].