Here $\varphi=\frac{1+\sqrt{5}}{2}$. It's true that $\mathbb{Z}[\varphi]=\mathcal{O}_{\mathbb{Q}(\sqrt{5})}$ is Euclidean since $\mathbb{Q}(\sqrt{5})$ is norm Euclidean, and I've read that $A=\mathbb{Z}\left[\frac{1+i}{\sqrt{2}}\right]$ is Euclidean as well, though I'm not certain what the Euclidean function is there (the reasonable candidate being $x\mapsto N_{K/\mathbb{Q}}(x)$, $K$ being $A$'s fraction field). So, my questions are:
- Is $R=\mathbb{Z}[i,\varphi]$ Euclidean?
- If so, what is the Euclidean function?
- The reasonable candidates I can think of are $x\mapsto x\bar{x}$ and $x\mapsto N_{L/\mathbb{Q}}(x)$ ($L$ being $R$'s fraction field), but these seem hard to work with for a proof.
- If not, is $R$ a (finite-index) subring of a (nice) Euclidean domain?
- The other fact I know about $R$ is that $R=\mathcal{O}_L$ by computing its discriminant and comparing with $D_L$ given by LMFDB, so $R$ is a PID.
- My main goal is to compute $\gcd$s in $R$, so if all of the above don't have affirmative answers, a Euclidean algorithm in $R$ would be great instead.
Thanks in advance for any answers.