Euclidean not implies norm-Euclidean - Revision history

Vipul: 3 revisions

2008-05-12T16:20:15Z

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Vipul: /* References */

2008-01-05T19:14:23Z

References

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	==References==		==References==

			* ''The Euclidean algorithm for Galois extensions of the rational numbers'' by David A. Clark
			* ''The Euclidean algorithm in Galois extensions of <math>\mathbb{Q}</math>'' 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)''		* ''A Quadratic Field which is Euclidean but not norm-Euclidean'' by David A. Clark, ''manuscripta math. 83, 327-330 (1994)''

Vipul: /* Example */

2008-01-05T19:11:35Z

Example

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	The ring:		The ring:

	<math>\mathbb{Z}\left[\frac{1 + \sqrt{69}{2}\right]</math>		<math>\mathbb{Z}\left[\frac{1 + \sqrt{69}}{2}\right]</math>

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

Vipul at 19:10, 5 January 2008

2008-01-05T19:10:12Z

New page

==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:

<math>\mathbb{Z}\left[\frac{1 + \sqrt{69}{2}\right]</math>

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

==Proof (for example)==

==References==

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