
Review of Short Phrases and Links 
This Review contains major "Euclidean Domain" related terms, short phrases and links grouped together in the form of Encyclopedia article.
Definitions
 A Euclidean domain is an integral domain which can be endowed with at least one Euclidean function.
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 A Euclidean domain is always a principal ideal domain (PID), an integral domain in which every ideal is a principal ideal.
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 Any Euclidean domain is a unique factorization domain (UFD), although the converse is not true.
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 Also, any Euclidean domain is a PID; the algorithm used to calculate greatest common divisors may be used to find a generator of any ideal.
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 Any Euclidean domain is a unique factorization domain (UFD), although the converse is not true.
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 A Euclidean domain is an integral domain which can be endowed with at least one Euclidean function.
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 Again, the converse is not true: not every PID is a Euclidean domain.
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 Also, any Euclidean domain is a PID; the algorithm used to calculate greatest common divisors may be used to find a generator of any ideal.
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 However, not every PID is a Euclidean domain; the ring furnishes a counterexample.
 In abstract algebra, a Euclidean domain (also called a Euclidean ring) is a type of ring in which the Euclidean algorithm can be used.
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 If R is a Euclidean domain then a form of the Euclidean algorithm can be used to compute greatest common divisors.
 Summary: Euclidean domain = principal ideal domain = unique factorization domain = integral domain = Commutative ring.
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 Despite this counterexample, the polynomial ring over any field is always a principal ideal domain and in fact, a Euclidean domain.
Euclidean Domain
 A Euclidean domain is always a principal ideal domain (PID), an integral domain in which every ideal is a principal ideal.
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 This is called "division with remainder" or " polynomial long division " and shows that the ring F[ X] is a Euclidean domain.
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 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.
Categories
 Principal Ideal Domain
 Euclidean Algorithm
 Pid
 Integral Domain
 Counterexample

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