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.
Read more about Euclidean Domain: Examples, Properties, Norm-Euclidean Fields
Famous quotes containing the word domain:
“The vice named surrealism is the immoderate and impassioned use of the stupefacient image or rather of the uncontrolled provocation of the image for its own sake and for the element of unpredictable perturbation and of metamorphosis which it introduces into the domain of representation; for each image on each occasion forces you to revise the entire Universe.”
—Louis Aragon (1897–1982)