Norm-Euclidean ring of integers: Difference between revisions

Latest revision as of 20:26, 26 January 2009

This article defines a property that can be evaluated for a ring of integers in a number field

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.

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

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

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 ==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.
+===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]].
 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==
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 ===Weaker properties===
-* [[Euclidean ring of integers]]
+* [[Stronger than::Euclidean ring of integers]]: {{proofofstrictimplicationat|[[Norm-Euclidean implies Euclidean]]|[[Euclidean not implies norm-Euclidean]]}}