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