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Integral Domain Polynomial Ring Ring Field U F D Gcd Domain Euclids Algorithm P I D Euclidean Valuation

1	Polynomial Ring
1	Field
1	Integral Domain
1	Ring
2	U F D
3	Euclidean Valuation
3	P I D
4	Gcd Domain
4	Euclids Algorithm

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Euclidean Ring

A Euclidean domain is an integral domain on which a Euclidean valuation can be defined.

Every Euclidean domain is a principal ideal domain, and therefore also a unique factorization domain.

Any two elements of a Euclidean domain have a greatest common divisor, which can be computed using the Euclidean algorithm.

An example of a Euclidean domain is the ring $\mathbb{Z}$ . Another example is the polynomial ring , where is any field. Every field is also a Euclidean domain.