Prime decomposition in number rings
We have seen that number rings are not always unique factorization domains: Elements may not factor uniquely into irreducibles. (See exercise 29, chapter 1, and exercise 15, chapter 2 for examples of non-unique factorization.) However we will prove that the nonzero ideals in a number ring always factor uniquely into prune ideals. This can “be regarded as a generalization of unique factorization in ℤ, where the ideals are just the principal ideals (n) and the prime ideals are the ideals (p), where p is a prime integer.
KeywordsPrime Ideal Number Field Principal Ideal Free Abelian Group Monic Polynomial
Unable to display preview. Download preview PDF.