I'm trying to prepare an exercise about rationalisation, which can be "simplified" at the end.
Something of the form $$\frac{p+q\sqrt d}{r+s\sqrt d}=\frac{pr-dqs+(qr-ps)\sqrt d}{r^2-ds^2}$$ with $p,q,r,s,d\in\Bbb Z\setminus\{0\}$ and $\sqrt d$ real and irrational.
I want that $\gcd(r^2-ds^2,pr-dqs,qr-ps)>1$. But I also want that $\gcd(r,s)=1$, to "hide" this simplification at the beginning.
I suspect that $\Bbb Z[\sqrt d]$ can't be an UFD, but I'm not sure.
I'll appreciate any help, but I prefer a method to a single, "magic" example.
Consider $$(a+b \sqrt{d})(a'+b' \sqrt{d})=(aa'+dbb') + (ab'+a'b) \sqrt{d}$$ and then divide by one of the factors.
For example $$(3-2 \sqrt{5})(1+ \sqrt{5}) = -7+ \sqrt{5}$$ means that $$\frac{-7+ \sqrt{5}}{3-2 \sqrt{5}} = 1+ \sqrt{5}$$
There is no magic behind it. It's just like when you prepare exercizes on quadratic formula and the discriminant magically appears to be a square: that's because you choose two rational roots $a,b$ and then construct the polynomial with those two roots simply expanding $(x-a)(x-b)$.