1. This led to modern abstract algebraic notions such as Euclidean domains . 2. The unique factorization of Euclidean domains is useful in many applications. 3. The quadratic integer rings are helpful to illustrate Euclidean domains . 4. The rings for which such a theorem exists are called Euclidean domains . 5. This algorithm and the associated proof may also be extended to any Euclidean domain . 6. Again, the converse is not true : not every PID is a Euclidean domain . 7. All Euclidean domains are principal ideal domains. 8. Any Euclidean domain is a unique factorization domain ( UFD ), although the converse is not true. 9. Important examples are polynomial rings over the integers or over a field, Euclidean domains and principal ideal domains. 10. Examples of Euclidean domains include fields, polynomial rings in one variable over a field, and the Gaussian integers.
