Finding GCD of two elements over a quadratic extension of integers

Question

I'm asking for a step-by-step explanation on finding $GCD(1+\sqrt{13}, 5+2\sqrt{13})$ in $\mathbb{Z}\left[\frac{1+\sqrt{13}}{2}\right]$.

Answer 1

Note that $\Bbb{Z}[\frac{1+\sqrt{13}}{2}]$ is Norm Euclidean (see this post and the references given there), so we can apply the Euclidean algorithm to find the GCD. $$\begin{align} 5+2\sqrt{13}&= 2(1+\sqrt{13}) +3 \\ 1+\sqrt{13}&= -1\cdot 3 +(4+\sqrt{13}) \\ 3 &= (4-\sqrt{13})(4+\sqrt{13}) + 0 \end{align}$$ So we see $4+\sqrt{13}$ is the GCD (up to associates) and $\dfrac{1+\sqrt{13}}{4+\sqrt{13}}=-3+\sqrt{13}$ and $\dfrac{5+2\sqrt{13}}{4+\sqrt{13}}=-2+\sqrt{13}$