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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 ...
1 Examples. 2 References. 3 Notes. ... In mathematics, a noncommutative unique factorization domain is a noncommutative ring with the unique factorization property.
As the Gaussian integers form a principal ideal domain, they also form a unique factorization domain. This implies that a Gaussian integer is irreducible (that is, it is not the product of two non-units) if and only if it is prime (that is, it generates a prime ideal). The prime elements of Z[i] are also known as Gaussian primes. An associate ...
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 ...
There is a unique ring homomorphism φ from Z[α] to Z/nZ that maps α to m. For simplicity, we'll assume that Z[α] is a unique factorization domain; the algorithm can be modified to work when it isn't, but then there are some additional complications. Next, we set up two parallel factor bases, one in Z[α] and one in Z.
As every polynomial ring over a field is a unique factorization domain, every monic polynomial over a finite field may be factored in a unique way (up to the order of the factors) into a product of irreducible monic polynomials. There are efficient algorithms for testing polynomial irreducibility and factoring polynomials over finite fields.
In mathematics and computer algebra the factorization of a polynomial consists of decomposing it into a product of irreducible factors.This decomposition is theoretically possible and is unique for polynomials with coefficients in any field, but rather strong restrictions on the field of the coefficients are needed to allow the computation of the factorization by means of an algorithm.
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 .