Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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.
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook




Loading...
Similar Threads for Norms Units Integral
I Finite Integral Domains ... Adkins & Weintraub, Propn 1.5