Euclidean not implies norm-Euclidean: Difference between revisions
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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. | ||
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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)'' | ||
Latest revision as of 16:20, 12 May 2008
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:
![{\displaystyle \mathbb {Z} \left[{\frac {1+{\sqrt {69}}}{2}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/02d4b770f84352f8bb6a9eb398f15a0661627017)
is a Euclidean domain, but is not norm-Euclidean.
Proof (for example)
References
- The Euclidean algorithm for Galois extensions of the rational numbers by David A. Clark
- The Euclidean algorithm in Galois extensions of 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)
