How to show that the ring of integers of $\mathbb{Q}(\sqrt{5})$, i.e., $\mathbb{Z}[\phi]$ is a principal ideal domain (where $\phi$ is the golden ratio)?
I want to prove that using as elementary algebra-level notions as possible.
Give some advice! Thank you!
The ring of integers of $\Bbb Q(\sqrt{5})$ is Euclidean, and hence a PID. The proof is by elementary algebra.
Reference: Prove that $R =\mathbb{Z}\left[\frac{1+\sqrt{5}}{2}\right]$ is a Euclidean domain.
The details of the proof are very similar to the ones of this post:
Proof that $\mathbb Z[\sqrt{3}]$ is a Euclidean Domain