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Factorization Problem

  1. May 21, 2006 #1
    I need to determine whether or not [itex]\mathcal{O}_{-10} = \mathbb{Z}[\sqrt{-10}][/itex] is a unique factorization domain.

    Now, I think the short answer is simply: NO.

    The question is meant to be simple (I think).

    I just finished proving that [itex]\mathcal{O}_{-5} = \mathbb{Z}[\sqrt{-5}][/itex] is NOT a unique factorization domain and it took me two pages. It involved me finding an integer which had two DIFFERENT factorizations into irreducibles. Using maple and the "factorEQ" command with (numtheory) I found that

    [tex]21 = 3\cdot 7 = (1+2\sqrt{-5})(1-2\sqrt{-5})[/itex]

    But now for the question at hand, Maple cannot find an integer which equals two factorizations of this sort because, unlike 5, 10 is not a prime.

    Ill let you guys muse over this for a while.
     
    Last edited: May 21, 2006
  2. jcsd
  3. May 21, 2006 #2
    I suppose I could just use

    [tex]26 = 2\cdot 13 = (4+\sqrt{-10})(4-\sqrt{-10})[/tex]

    and show that no element of [itex]\mathbb{Z}[\sqrt{-10}][/itex] of norm 2 or 13 hence 2 and 13 are irreducible. And then show there is no element of norm [itex]4+\sqrt{-10}[/itex] or [itex]4-\sqrt{-10}[/itex] and thus, they are irreducible. If I can show that then I have shown that 26can be written as two DIFFERENT factorizaruib into irreducibles which implies the ring is a UFD. This shouldnt be too hard if its the right thing to do. Is it the right thing to do?
     
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