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).