Example 4

\(z=-1+4\sqrt{2}\;\;\;\;\;\;\;\;\;\; w=\sqrt{2}\)

\(\begin{array}{rcl} -1+4\sqrt{2} & = & \sqrt{2}\left(3-\sqrt{2}\right)+1+\sqrt{2}\\ \sqrt{2} & = & \left(1+\sqrt{2}\right)\left(2-\sqrt{2}\right) \end{array}\)

\(gcd\left(-1+4\sqrt{2},\sqrt{2}\right)=1+\sqrt{2}\)