Norm-Euclidean ring of integers: Difference between revisions
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==Definition== | ==Definition== | ||
The [[ring of integers]] of a [[number field]] is termed '''norm-Euclidean''' if | The [[ring of integers]] of a [[number field]] is termed '''norm-Euclidean''' if the absolute value of the [[algebraic norm in a ring of integers|algebraic norm]] is a [[Euclidean norm]]. | ||
Since the norm in a ring of integers is multiplicative, norm-Euclidean rings possess [[multiplicative Euclidean norm]]s. | Since the norm in a ring of integers is multiplicative, norm-Euclidean rings possess [[multiplicative Euclidean norm]]s. | ||
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===Weaker properties=== | ===Weaker properties=== | ||
* [[Euclidean ring of integers]] | * [[Stronger than::Euclidean ring of integers]]: {{proofofstrictimplicationat|[[Norm-Euclidean implies Euclidean]]|[[Euclidean not implies norm-Euclidean]]}} | ||
Revision as of 02:28, 24 January 2009
This article defines a property that can be evaluated for a ring of integers in a number field
Definition
The ring of integers of a number field is termed norm-Euclidean if the absolute value of the algebraic norm is a Euclidean norm.
Since the norm in a ring of integers is multiplicative, norm-Euclidean rings possess multiplicative Euclidean norms.
Relation with other properties
Weaker properties
- Euclidean ring of integers: For proof of the implication, refer Norm-Euclidean implies Euclidean and for proof of its strictness (i.e. the reverse implication being false) refer Euclidean not implies norm-Euclidean