Algebraic Norm/Conjugates

Define $\mathbb{Z}[\sqrt{n}]$ to be the set $\{a+b\sqrt{n}:a,b\in\mathbb{Z}\}$.

Consider $\alpha=a+b\sqrt{n}\in\mathbb{Z}[\sqrt{n}]$. We define the conjugate of $\alpha$ to be $\overline{\alpha}:=a-b\sqrt{n}$. Then, we naturally define the norm as $N(\alpha)=\alpha\overline\alpha=a^2-nb^2$.

Note that if $n=-1$, we have $N(\alpha)=a^2+b^2$, which is $|\alpha|^2$.

The way we defined norms makes it easy to check that the norm is multiplicative. $$N(\alpha\beta) = \alpha\beta\overline{\alpha\beta} = \alpha\overline{\alpha}\beta\overline\beta = N(\alpha)N(\beta).$$

Let's try some examples: