Consider the order $R=\mathbb{Z}(\sqrt[4]{-19}) \subset K=\mathbb{Q}(\sqrt[4]{-19}) $. Let $\alpha=\sqrt[4]{-19}$ and $\beta=(\alpha^3+\alpha^2+\alpha+1)/2$. My question is:
How do we compute the index of $R \subset R[\beta]$?
The discriminant $\Delta(R)=\Delta(X^4+19)=2^8\cdot 19^3$ and $\Delta(R) = [R[\beta]:R]^2 \cdot \Delta(R[\beta])$. Therefore if we can compute the discriminant of $R[\beta]$ then we get the index. However i find $\Delta(R[\beta])$ so complicated. Maybe we have another approach to know $[R[\beta]:R]$ then can figure out $\Delta(R[\beta])$.