Norms and Units of an Integral Domain

  • Thread starter alexfloo
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  • #1
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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:

Therefore, if [itex]z=a+\sqrt{-5}b[/itex] is a unit, then [itex]N(z)=a^2+5b^2=1[/itex].
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).
 

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  • #2
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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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