I am reading about how one can use the Dedekind-Hasse norm to show that $\textbf{Z}[(1+\sqrt{-19})/2]$ is a Principal Ideal Domain. I understand the proof outlined in the link I gave, but I have a question about step C. of part ii. at the bottom of page 3. They essentially define a bunch of new variables, and it indeed turns out that such definitions do work out in the proof, which I can understand. However, I don't really know how I would be able to come up with something like that; it really does seem like they just pulled these out of nowhere.
I ended up deriving that $$N\left(\frac\alpha\beta s-t\right)=\frac{a^2s^2-2acst+19b^2+c^2t^2}{c^2}=\frac{(as-ct)^2+19b^2}{c^2}$$ but I am not sure how to proceed to what they ended up getting. Can someone shed light on this?
we have $\frac{\alpha}{\beta}=\frac{a+b\delta}{c}$, $s=y+x\delta$, $t=q-z\delta$, $ax+by+cz=1$, and $ay+\delta^2 bx=cq+r$, where $\delta=\sqrt{-19}$.
Now \begin{align*} \frac{\alpha}{\beta}s-t& = \left(\frac{a+b\delta}{c}\right)(y+x\delta)-q+z\delta\\ &=\left(\frac{ay+bx\delta^2}{c}-q\right)+\left(\frac{ax+by}{c}+z\right)\delta\\ &=\left(\frac{cq+r}{c}-q\right)+\left(\frac{1-cz}{c}+z\right)\delta\\ &=\left(\frac{r}{c}\right)+\left(\frac{1}{c}\right)\delta. \end{align*} Now $$N\left(\frac{\alpha}{\beta}s-t\right)=\left(\frac{r}{c}\right)^2+19\left(\frac{1}{c}\right)^2=\frac{r^2+19}{c^2}$$