Computing the index $\left(\mathbb Z\left[\frac{1+\sqrt{5}}{2}\right]:\mathbb Z \left[\sqrt{5}\right]\right)$?

Let $\theta=\frac{1+\sqrt{5}}{2}$. Then $\mathbb Z[\sqrt{5}]=\mathbb Z 1 + \mathbb Z 2\theta$ and $\mathcal O = \mathbb Z 1 + \mathbb Z \theta$. Therefore, $\left(\mathcal O:\mathbb Z [\sqrt{5}]\right)= 2$.

Equivalently, write $$ \pmatrix{1 \\ \sqrt 5} = \pmatrix{\hphantom{-}1 & 0 \\ -1 & 2} \pmatrix{1 \\ \frac{1+\sqrt{5}}{2}} $$ and note that the determinant of the matrix is $2$.