Euclidean Domain

A Euclidean domain is an integral domain which can be endowed with at least one Euclidean function. It is important to note that a particular Euclidean function f is not part of the structure of a Euclidean domain: in general, a Euclidean domain will admit many different Euclidean functions.

Most algebra texts require a Euclidean function to have the following additional property:

• (EF2) For all nonzero a and b in R, f(a) ≤ f(ab).

However, one can show that (EF2) is superfluous in the following sense: any domain R which can be endowed with a function g satisfying (EF1) can also be endowed with a function f satisfying (EF1) and (EF2): indeed, for one can define f(a) as follows

In words, one may define f(a) to be the minimum value attained by g on the set of all non-zero elements of the principal ideal generated by a.