Euclidean Domain and Divison Algorithm

Asked Feb 16 '15 at 17:43

Active Feb 16 '15 at 19:06

Viewed 45 times

In ED definition we need for all $a, b$ not zero then there exist $q$ and $r$ such that $a = bq+r$ with some condition with respect to norm.

Actually I am interested whether $q$ and $r$ are unique in ED or not

Further Questions: 1. Can we define some norm in ED with respect to which q and r are unique And in how many ways we can define such norms

If it is not possible what algebraic structure like this i.e. ED with uniqueness of q and r, has some special properties.

edited Feb 16 '15 at 18:34

asked Feb 16 '15 at 17:43

Sushil

Yes exactly because of uniqueness in k[x] i asked this question – Sushil Feb 16 '15 at 18:18
Uniqueness holds only for fields $,F,$ and univariate polynomial rings over fields $,F[x],,$ see the links to papers of Jodeit and Rhai in this answer, which gives a nce application of such uniqueness. – Bill Dubuque Feb 16 '15 at 19:02

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