Vipul: New page: ==Statement== ===Verbal statement=== Any fact about::Euclidean norm on a commutative unital ring is a fact about::Dedekind-Hasse norm. ==Definitions used== ===Euclidean nor...

2009-01-23T18:32:27Z

New page: ==Statement== ===Verbal statement=== Any fact about::Euclidean norm on a commutative unital ring is a fact about::Dedekind-Hasse norm. ==Definitions used== ===Euclidean nor...

New page

==Statement==

===Verbal statement===

Any [[fact about::Euclidean norm]] on a [[commutative unital ring]] is a [[fact about::Dedekind-Hasse norm]].

==Definitions used==

===Euclidean norm===

{{further|[[Euclidean norm]]}}

===Dedekind-Hasse norm===

{{further|[[Dedekind-Hasse norm]]}}

==Related facts==

* [[Dedekind-Hasse not implies Euclidean]]

==Proof==

===Key idea===

The main idea here is that while the Euclidean condition allows us to find an element of the form <math>a - bq</math> of smaller norm, the Dedekind-Hasse condition only requires some <math>R</math>-linear combination of <math>a</math> and <math>b</math> to have smaller norm. In particular, for the Dedekind-Hasse condition, we can pick <math>sa - bq</math> with <math>s \ne 1</math>.

===Proof details===
'''Given''': A commutative unital ring <math>R</math> with a Euclidean norm <math>N</math>.

'''To prove''': <math>N</math> is a Dedekind-Hasse norm on <math>R</math>: given <math>a,b \in R</math> with <math>b \ne 0</math>, either <math>a \in (b)</math> or there exists <math>r \in (a,b)</math> such that <math>N(r) < N(b)</math>.

'''Proof''': Since <math>b \ne 0</math> and <math>N</math> is a Euclidean norm, we can write:

<math>a = bq + r</math>

where either <math>r = 0</math> or <math>N(r) < N(b)</math>. Consider both cases:

* <math>r = 0</math>: In this case, <math>a = bq</math>, so <math>a \in (b)</math>, so the Dedekind-Hasse condition is satisfied.
* <math>N(r) < N(b)</math>: In this case, <math>r = a - bq</math>, so <math>r \in (a,b)</math>, with <math>N(r) < N(b)</math>, so the Dedekind-Hasse condition is satisfied.

Euclidean implies Dedekind-Hasse - Revision history

Vipul: New page: ==Statement== ===Verbal statement=== Any fact about::Euclidean norm on a commutative unital ring is a fact about::Dedekind-Hasse norm. ==Definitions used== ===Euclidean nor...