What is the norm and how do you calculate it in this case?

Question

I am reading the following proof that 5 is prime in $\mathbb{Z}[\sqrt{2}]$:

Suppose that $5\mid (a+b\sqrt{2})(c+d\sqrt{2})$ for $a+b\sqrt{2},c+d\sqrt{2}$. By taking the norm, we obtain $25\mid (a^2-2b^2)(c^2-2d^2)$ in $\mathbb{Z}$

My question is, what exactly is a norm and how is it calculated?

Answer 1

If $\alpha=a+b\sqrt{2}\in\mathbb{Z}[\sqrt{2}]$, then the norm of $\alpha$ is $$N(\alpha)=(a+b\sqrt{2})(a-b\sqrt{2})=a^2 - 2b^2$$

The norm $N$ is just a useful function $$N:\mathbb{Z}[\sqrt{2}]\rightarrow \mathbb{N}$$that has some nice properties:

• $N(\alpha)=1\iff\alpha$ is a unit
• $N(\alpha)N(\beta)=N(\alpha\beta)$