Norm (Quadratic Integers)

definitionnumber-theory

Definition

For any $\alpha \in \mathbb{Z}[\sqrt{d}]$ with $d\in \mathbb{Z}$, the norm of $\alpha$, $N(\alpha )$, is equal to

$$N(\alpha )=a^2-d{b}^2.$$

Note

For $d<0$, $N(\alpha )$ can only be positive.