Euclidean domain

The proof that follows may be adapted for any Euclidean domain.

The rings for which such a theorem exists are called Euclidean domains.

The unique factorization of Euclidean domains is useful in many applications.

Robert Seeley extended this to include certain Euclidean domains in 1978.

Euclidean domains appear in the following chain of class inclusions:

Again, in the language of abstract algebra, the above says that Z is a Euclidean domain.

All Euclidean domains are principal ideal domains, but the converse is not true.

All Euclidean domains and all fields are principal ideal domains.

He was the first to prove the existence of principal ideal domains that are not Euclidean domains, being his first example.

Examples of Euclidean domains include:

Euclidean domains are integral domains in which the Euclidean algorithm can be carried out.

Moreover, one can define an operation division with remainder, which makes the Dirichlet integers into a Euclidean domain.

A Euclidean domain is an integral domain in which a degree function is defined so that "division with remainder" can be carried out.

However, there are quadratic integer rings that are principal ideal domains but not Euclidean domains.

A Euclidean domain is an integral domain which can be endowed with at least one Euclidean function.

Secondly, it is very similar to the case of the integers, and this analogy is the source of the notion of Euclidean domain.

Again, the converse is not true: not every PID is a Euclidean domain.

In Euclidean domain, many authors use "norm function" in place of "Euclidean function".

It can be shown that the degree of a polynomial over a field satisfies all of the requirements of the norm function in the euclidean domain.

This generalization assumes that the set of periodic billiards has measure 0, which Ivrii conjectured is fulfilled for all bounded Euclidean domains with smooth boundaries.

It is important to compare the class of Euclidean domains with the larger class of principal ideal domains (PIDs).

Strictly speaking it is the ring of integers that is Euclidean since fields are trivially Euclidean domains, but the terminology is standard.

Since the ring of polynomials over a field is an Euclidean domain, we may compute these GCDs using the Euclidean algorithm.

The third condition is a slight generalisation of condition (EF1) of Euclidean functions, as defined on the Euclidean domain article.

(This does not however mean the ordinals are a Euclidean domain, since they are not even a ring, and the Euclidean "norm" is ordinal-valued.)