# Euclidean domains

this seems to be a very fundamental problem...but i need help...
prove or disprove : let D be a euclidean domain with size function d, then for a,b in D, b != 0, there exist unique q,r in D such that a= qb+r where r=0 or d(r) < d(b).
first of all, what is size function? next...do we only need to show the existence of unique q and r?

matt grime
Homework Helper
I suspect that size function means exactly that - it is a well ordering on the set D, a way of comparing two elements for "size", though you ought to read the place where this appears as it will define it properly (I don't think it's standard, in that I've never heard of it, but that means nothing, I suppose).

I'd imagine that this means given x in D and a set of elements, Y, then you may pick the largest element of Y that is less than x.

There will also be some linearity condition, ie, d(x+y)=d(x)+d(y), and that d(a)<d(b) implies d(a)+d(x) < d(b)+d(x)

mathwonk
Homework Helper
2020 Award
since you are given that it is a euclidean domain, the concept of size function is part of the definition. I.e. look where they define eucldiean domain, and the size function will be defined.

For example in my book it is as follows:

Definition: A domain is called a Euclidean domain if the division algorithm holds in the following form: to each non zero element a of R there is associated a non negative integer d(a), such that
(i) for a,b non zero in R, we have d(a)<=d(ab),
(ii) for a,b non zero in R, there exist t,r in R such that a = bt +r, and either r=0 or d(r) < d(b). [Note: uniqueness of t,r, is not required.]

this revelas a likely meaning of your problem. i.e. they probably have defined euclidean domain as i have and merely asking you about uniqueness of the t,r or in your notation the q,r.