It is important to compare the class of Euclidean domains with the larger class of principal ideal domains ( PIDs ).
12.
B�zout's identity, and therefore the previous algorithm, can both be generalized to the context of Euclidean domains.
13.
Since the ring of polynomials over a field is a Euclidean domain, we may compute these GCDs using the Euclidean algorithm.
14.
Since the ring of polynomials over a field is an Euclidean domain, we may compute these GCDs using the Euclidean algorithm.
15.
A "'Euclidean domain "'is an integral domain which can be endowed with at least one Euclidean function.
16.
Strictly speaking it is the ring of integers that is Euclidean since fields are trivially Euclidean domains, but the terminology is standard.
17.
The third condition is a slight generalisation of condition ( EF1 ) of Euclidean functions, as defined in the Euclidean domain article.
18.
The fundamental theorem of arithmetic applies to any Euclidean domain : Any number from a Euclidean domain can be factored uniquely into irreducible elements.
19.
The fundamental theorem of arithmetic applies to any Euclidean domain : Any number from a Euclidean domain can be factored uniquely into irreducible elements.
20.
A Euclidean domain is always a principal ideal domain ( PID ), an integral domain in which every ideal is a principal ideal.
