Euclidean not implies norm-Euclidean

Statement

A ring of integers in a number field may be a Euclidean domain, even though it is not a norm-Euclidean domain. In other words, it may have a Euclidean norm which differs from its norm function, even if it is not Euclidean under its norm function.

Example

The ring:

$\mathbb{Z}[\frac{1+\sqrt{69}}{2}]$

is a Euclidean domain, but is not norm-Euclidean.

References

• A Quadratic Field which is Euclidean but not norm-Euclidean by David A. Clark, manuscripta math. 83, 327-330 (1994)