- #1
alexfloo
- 192
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In an example in my algebra text, (from the section on unique factorization domains) it is describing the ring \mathbb{Z}[\sqrt{-5}], and demonstrating that it is not a UFD. It starts by giving the norm
N(a+\sqrt{-5}b)=a^2+5b^2.
It remarks that if zw=1, then N(z)N(w)=1, 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 \mathbb{Z}[\sqrt{-5}], which would then be units (which is what he appears to be assuming).
N(a+\sqrt{-5}b)=a^2+5b^2.
It remarks that if zw=1, then N(z)N(w)=1, 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 \mathbb{Z}[\sqrt{-5}], which would then be units (which is what he appears to be assuming).
