Proof a number is Fibonacci number I have a question regarding the proof that a number n is a Fibonacci number if and only if $5n^2-4$ or $5n^2+4$ is a perfect square. I don't understand the second part of the proof: knowing that $5n^2-4$ or $5n^2+4$ is a perfect square, prove that n is a Fibonacci number. 
I attach the solution I found in this Google Book - Fibonacci and Lucas Numbers with Applications by Thomas Koshy:

I don't understand the part of the proof starting from "Since m and n...". 
Basically:


*

*Why must $(m + n \sqrt5)/2$ and $(m - n \sqrt5)/2$ be integers in the given extension field?

*Why must they be units in this field if their product is -1 (what is actually "a unit in an extension field" and why does it have that form?)


Thank you in advance.

The ring of integers $\mathcal{O}_K$ in the quadratic number field $K=\mathbb{Q}(\sqrt{5})$ is given by $\mathbb{Z}+\frac{1+\sqrt{5}}{2}\mathbb{Z}$. This is a basic result of algebraic number theory. So the author means that these elements are in the ring of integers of the extension field. Furthermore the ring $\mathcal{O}_K$ has a unit group, i.e., its invertible elements. The product equal $\pm 1$ equation says that they are invertible. Finally we use a famous theorem on the unit group of number fields, i.e., Dirichlet's unit theorem. It says in our case that the unit group of $\mathcal{O}_K$ is infinite cyclic times $\{\pm 1\}$.
