Norms and Units of an Integral Domain

[alexfloo]

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:

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

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).

 

[alexfloo]

Nevermind, I see now. We can't have reciprocal complex elements because the smallest nonzero value the norm can have is 1.