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Formally, a unique factorization domain is defined to be an integral domain R in which every non-zero element x of R which is not a unit can be written as a finite product of irreducible elements p i of R: x = p 1 p 2 ⋅⋅⋅ p n with n ≥ 1. and this representation is unique in the following sense: If q 1, ..., q m are irreducible elements ...
Thus, a number field has class number 1 if and only if its ring of integers is a principal ideal domain (and thus a unique factorization domain). The fundamental theorem of arithmetic says that Q has class number 1.
This complete factorization is unique up to the order of the factors. If the coefficients of P(x) are real, one generally wants a factorization where factors have real coefficients. In this case, the complete factorization may have some quadratic (degree two) factors. This factorization may easily be deduced from the above complete factorization.
Many more authors state theorems for Dedekind domains with the implicit proviso that they may require trivial modifications for the case of fields. An immediate consequence of the definition is that every principal ideal domain (PID) is a Dedekind domain. In fact a Dedekind domain is a unique factorization domain (UFD) if and only if it is a PID.
Then the factorization problem is reduced to factorize separately the content and the primitive part. Content and primitive part may be generalized to polynomials over the rational numbers, and, more generally, to polynomials over the field of fractions of a unique factorization domain.
In the case of coefficients in a unique factorization domain R, "rational numbers" must be replaced by "field of fractions of R". This implies that, if R is either a field, the ring of integers, or a unique factorization domain, then every polynomial ring (in one or several indeterminates) over R is a unique factorization domain. Another ...
Multiplication is defined for ideals, and the rings in which they have unique factorization are called Dedekind domains. There is a version of unique factorization for ordinals, though it requires some additional conditions to ensure uniqueness. Any commutative Möbius monoid satisfies a unique factorization theorem and thus possesses ...
The converse is true for unique factorization domains [2] (or, more generally, GCD domains). Moreover, while an ideal generated by a prime element is a prime ideal , it is not true in general that an ideal generated by an irreducible element is an irreducible ideal .