Following my other thread: Finding units of $\mathbb{Z}[\delta]$ with $\delta=\frac{1+\sqrt{d}}{2}$ and $d\equiv 1\pmod{4}.$
In the ring $\mathbb{Z}[\delta]:=\left\{a+b\delta:a,b\in\mathbb{Z}\right\}$ where $\delta:=\frac{1+\sqrt{d}}{2}$, $d$ be a square free integer, $d<0$ and $d\equiv 1\pmod{4}$. Let $N:\mathbb{Z}[\delta]\to \mathbb{N}_{0}$ norm, where $N(a+b\delta)=(a+b\delta)(a+b\overline{\delta})=a^2+ab+b^2\delta\overline{\delta}$.
Is there a way (more elegant) to prove that $N(\alpha\beta)=N(\alpha)N(\beta)$ for all $\alpha,\beta\in\mathbb{Z}[\delta]$ without using brute force calculation?