Abstract
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.
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© 1977 Springer-Verlag, New York Inc.
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Marcus, D.A. (1977). Prime decomposition in number rings. In: Number Fields. Universitext. Springer, New York, NY. https://doi.org/10.1007/978-1-4684-9356-6_3
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DOI: https://doi.org/10.1007/978-1-4684-9356-6_3
Publisher Name: Springer, New York, NY
Print ISBN: 978-0-387-90279-1
Online ISBN: 978-1-4684-9356-6
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