Question concerning the dedekind factoring of a principal ideal

Question

In this paper the author gives the criterium for the factoring of a principal ideal of the ring of integers of a number field. My problem is how $[\mathcal{O}_K:\Bbb Z[\alpha]]$ is calculated. I think I understand the meaning as it is the dimension of the quotient of a ring by a subring (as linear spaces). So I (presumably falsly) thought that it equals the determinant of a linear transformation mapping the integral basis to the basis consisting of the powers $1, \alpha, \alpha^2, \ldots$ . I now apply this to example 5 in the paper. I find an integral basis ${1, (3+\alpha+\alpha^2)/5,\alpha^2}$. Now the linear transformation that brings this basis to the first one is given by the matrix $$\begin{pmatrix}1&0&0\\-3&5&-1\\0&0&1\end{pmatrix}$$ with determinant $5$. Exactly not the prime(s) we were looking for. These are seemingly discr$(f)/5$.