Norm-Euclidean ring of integers - Revision history

Vipul at 20:26, 26 January 2009

2009-01-26T20:26:03Z

← Older revision		Revision as of 20:26, 26 January 2009
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	==Definition==		==Definition==

			===Definition in terms of Euclidean norms===

	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]].		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.

			===Definition in terms of norms in the field of fractions===

			The [[ring of integers]] <math>\mathcal{O}</math> of a [[number field]] <math>K</math> is termed '''norm-Euclidean''' if for any <math>x \in K</math>, there exists <math>y \in \mathcal{O}</math> such that <math>N(x - y) < 1</math>, where <math>N</math> denotes the [[algebraic norm in a number field]].

			===Equivalence of definitions===

			{{proofat\|[[Equivalence of definitions of norm-Euclidean ring of integers]]}}

	==Relation with other properties==		==Relation with other properties==

Vipul at 02:28, 24 January 2009

2009-01-24T02:28:54Z

← Older revision		Revision as of 02:28, 24 January 2009
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	==Definition==		==Definition==

	The [[ring of integers]] of a [[number field]] is termed '''norm-Euclidean''' if ~~it is a [[Euclidean domain]] where~~ the ~~norm is given by~~ the ~~usual~~ [[norm in a ring of integers]]~~. In other words, the~~ norm ~~is the product of all algebraic conjugates, counted to the correct multiplicity~~.		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]]}}

Vipul: 1 revision

2008-05-12T16:28:13Z

1 revision

← Older revision	Revision as of 16:28, 12 May 2008
(No difference)

Vipul at 23:39, 17 December 2007

2007-12-17T23:39:27Z

New page

{{ring of integers-property}}

==Definition==

The [[ring of integers]] of a [[number field]] is termed '''norm-Euclidean''' if it is a [[Euclidean domain]] where the norm is given by the usual [[norm in a ring of integers]]. In other words, the norm is the product of all algebraic conjugates, counted to the correct multiplicity.

Since the norm in a ring of integers is multiplicative, norm-Euclidean rings possess [[multiplicative Euclidean norm]]s.

==Relation with other properties==

===Weaker properties===

* [[Euclidean ring of integers]]