There is theorem that "Every principal ideal domain is a unique
factorization domain."
But in Fricas, "PrincipalIdealDomain has UniqueFactorizationDomain", or
"PID has UFD",
returns false.
In catdef.spad, we can find PID and UFD are both constructed from GcdDomain:
(comments removed)
PrincipalIdealDomain() : Category == GcdDomain with
principalIdeal : List % -> Record(coef : List %, generator : %)
expressIdealMember : (List %,%) -> Union(List %,"failed")
UniqueFactorizationDomain() : Category == GcdDomain with
prime? : % -> Boolean
squareFree : % -> Factored(%)
squareFreePart : % -> %
factor : % -> Factored(%)
And the only place where PID is used in Fricas is in EuclideanDomain:
EuclideanDomain() : Category == PrincipalIdealDomain with ....
The (only?) category constructed on EuclideanDomain is Field which includes
UFD:
Field() : Category == Join(EuclideanDomain, UniqueFactorizationDomain,
....
So, my question is, should "PID has UFD" returns true, aka PID buils upon
UFD
instead of GCDDOM?
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