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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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Could you explain what exactly you don't understand? Clearly, he covers all possible cases (presumably 2(b) should read $a\in \mathbb{Z}$ and $b\in 5\mathbb{Z}$). Case 1 covers $b\in \mathbb{Z}$, cases 2(a) and 2(b) together cover $a\in \mathbb{Z}$. So the remaining case is that neither is in $\mathbb{Z}$. For this case, cases 3(a) and 3(b) cover the case that either both are in $2\mathbb{Z}$ or neither is. Cases 3(c) and 3(d) the cover the remaining case that exactly one of them is in $2\mathbb{Z}$. –  Alex B. Nov 12 '11 at 14:30
    
What I don't understand is how he comes up with these cases. For, example why are there two different subcases when $a$ is an integer? or why there are four different subcases when when both $a$ and $b$ are not integers? And how should I know what those subcases should be? –  Mohan Nov 13 '11 at 6:34
    
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. –  Alex B. Nov 14 '11 at 0:20

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