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Unformatted text preview: 6.5. FACTORIZATION 297 ideal domain is a unique factorization domain. Thus we have the implica- tions: Euclidean Domain H) Principal Ideal Domain H) Unique Factorization Domain H) Integral Domain It is natural to ask whether any of these implications can be reversed. The answer is no. Some Examples Example 6.5.11. Z OE.1 C i p 19/=2Ł is a principal ideal domain that is not Euclidean. The proof of this is somewhat intricate and will not be presented here. See D. S. Dummit and R. M. Foote, Abstract Algebra , 2nd ed., Prentice Hall, 1999, pp. 278 and 283. Example 6.5.12. Z OExŁ is a unique factorization domain that is not a prin- cipal ideal domain. We will show later that if R is a unique factorization domain, then the polynomial ring ROExŁ is also a unique factorization do- main; see Theorem 6.6.7 . This implies that Z OExŁ is a UFD. Let’s check that the ideal 3 Z OExŁ C x Z OExŁ is not principal; this ideal is equal to the set of polynomials in Z OExŁ whose constant coefficient is a multiple of...
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This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.
- Fall '08