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Norms and Units of an Integral Domain

  1. Nov 9, 2011 #1
    In an example in my algebra text, (from the section on unique factorization domains) it is describing the ring [itex]\mathbb{Z}[\sqrt{-5}][/itex], and demonstrating that it is not a UFD. It starts by giving the norm

    [itex]N(a+\sqrt{-5}b)=a^2+5b^2[/itex].

    It remarks that if [itex]zw=1[/itex], then [itex]N(z)N(w)=1[/itex], and then it goes on immediately to say that:

    This certainly seems plausible, but I don't see that he's actually proved it. It's not evident that we could not have any pair of reciprocal complex elements of [itex]\mathbb{Z}[\sqrt{-5}][/itex], which would then be units (which is what he appears to be assuming).
     
  2. jcsd
  3. Nov 9, 2011 #2
    Nevermind, I see now. We can't have reciprocal complex elements because the smallest nonzero value the norm can have is 1.
     
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