A quadratic field which is Euclidean but not norm-Euclidean

TLDR: In this paper, the authors gave the first example for quadratic fields, the ring of integers of magnitude 69 for which the norm function is not Euclidean but not norm-Euclidean.

Abstract: The classification of rings of algebraic integers which are Euclidean (not necessarily for the norm function) is a major unsolved problem. Assuming the Generalized Riemann Hypothesis, Weinberger [7] showed in 1973 that for algebraic number fields containing infinitely many units the ring of integersR is a Euclidean domain if and only if it is a principal ideal domain. Since there are principal ideal domains which are not norm-Euclidean, there should exist examples of rings of algebraic integers which are Euclidean but not norm-Euclidean. In this paper, we give the first example for quadratic fields, the ring of integers of $$\mathbb{Q}\left( {\sqrt {69} } \right)$$ .