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:

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

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

References

• The Euclidean algorithm for Galois extensions of the rational numbers by David A. Clark
• The Euclidean algorithm in Galois extensions of $\mathbb {Q}$ by David A. Clark and M. R. Murty
• A Quadratic Field which is Euclidean but not norm-Euclidean by David A. Clark, manuscripta math. 83, 327-330 (1994)