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I was reading that the proof of the fact that $R =\mathbb Z [(1+\sqrt{-19})/2]$ is a principal ideal domain from here

It actually shows that $R$ is a Dedekind-Hasse domain, that is let $ \alpha , \beta \in R $ then there exists $ \gamma , \delta \in R$ such that $N(\alpha/\beta*\gamma-\delta) <1$, where $N$ is the D-H norm.

To prove that he comes up many cases. I am not able to understand how does he come up with these cases.

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    The general philosophy is: if you have an idea what to try, try it. Then realise that it only works in certain situations, not in general, so you are happy that you solved$a$special case of your problem and take it from there.2011-11-14

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