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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!

J. W. Tanner
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AnonyMath
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1 Answers1

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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

Dietrich Burde
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