Signal Processing on non-Euclidean domains

1. What is a non Euclidean domain?
2. Is every PID a Euclidean domain?
3. Is Z an Euclidean domain?
4. Is every principal ideal domain is Euclidean domain?

What is a non Euclidean domain?

In broad terms, non-Euclidean data is data whose underlying domain does not obey Euclidean distance as a metric between points in the domain.

Is every PID a Euclidean domain?

Theorem: Every Euclidean domain is a principal ideal domain. Proof: For any ideal , take a nonzero element of minimal norm .

Is Z an Euclidean domain?

The ring Z is a Euclidean domain. The function d is the absolute value. Definition 20.3.

Is every principal ideal domain is Euclidean domain?

It is well known that any Euclidean domain is a principal ideal domain, and that every principal ideal domain is a unique factorization domain. The main examples of Euclidean domains are the ring Z of integers and the polynomial ring K[x] in one variable x over a field K.