Factorization Domain In Algebra

Imagine a factorization domain where all irreducible elements are prime.

Factorization domain in algebra. Any element g x r x can be written as. Conversely if r is a ufd let an irreducible element p divide ab. Specifically a ufd is an integral domain in which every non zero non unit element can be written as a product of prime elements uniquely up to order and units.

Important examples of ufds are the integers and polynomial rings in one or more variables with coefficients coming from the integers or from a field. An integral domain r is a unique factorization domain ufd if every nonzero nonunit of r can be expressed as a product of irreducibles and furthermore the factorization is unique up to order and associates. In other words a factorization is an expression of a nonzero nonunit as a product of irreducible elements.

Norm functions an interesting link between number theory and algebra is afforded by the study of norm functions on rings namely on functions n. In mathematics a unique factorization domain is a ring in which a statement analogous to the fundamental theorem of arithmetic holds. Let r be a unique factorization domain and let f denote the field of fractions of r.

In a ufd all irreducibles are prime. We already know the prime elements are irreducible apply euclid s proof and the ring becomes a ufd. Since the factorization of ab is unique p appears somewhere in the factors of a or b hence p divides.

Call an element of r x primitive if its coefficients are relatively prime. Unique factorization domains appear in the following.