An exercise in my abstract-algebra class asks me that
Prove that $R =\mathbb{Z}\left[\frac{1+\sqrt{5}}{2}\right]$ is a Euclidean domain.
If $u$ is a unit of $R$ , then $$u=(\sqrt{5}+2)^n \;\;\text{or}\; \;u=-(\sqrt{5}+2)^n$$ for some $n \in \mathbb{Z}$.
Firstly I want to show that $R$ is a Euclidean domain with respect to the following function: $$\varphi: R\setminus\{0\} \to \mathbb{Z}_{+}: a+b\phi \mapsto |a^2 + ab - b^2|$$ where $\phi = \frac{1 + \sqrt{5}}{2}$.
But there's some problem in finishing them, any help or insight is deeply appreciated.
Edit：
To prove $R$ is a Euclidean we must to check that
- $\varphi(\alpha ) \leq \varphi(\alpha \beta )$ for $\beta \neq 0$;
- for any $\alpha, \beta \in R, \beta \neq 0$ ,there exist $q,r \in R$ such that $\alpha =q \beta +r$ with $r =0 $or $\varphi(r) < \varphi (\beta)$.
For $(1)$ obviously.
For $(2)$, we look for $q$ and $r$ such that $\alpha =q \beta +r$ and write this as $\frac{\alpha}{\beta}=q+\frac{r}{\beta}$, here $q,r \in \mathbb {Q}$. Then $$\frac{\alpha}{\beta}=\mu +\nu \frac{1+\sqrt{5}}{2}$$ for some $\mu ,\nu \in \mathbb {Q}$(since $\mathbb{Q}[\frac{1+\sqrt{5}}{2}]$ is a field .)However, I tried to find possible integers $m,n$ to approximate $\mu ,\nu$ to test $\varphi(r) < \varphi (\beta)$, but failed.
