I have found that the ring of integers is $\mathbb{Z}[\alpha, \alpha^3/4]$ where $\alpha = \sqrt[4]{24}$. I also know that in the ring of integers $(5)$ factors as two ideals of norm $25$, and $(2)$ factors as $(2, \alpha)^4$ hence there are 2 ideals of norm $100$. How does this help me find the ideals of norm $100$ in the smaller order (where there is no longer unique factorization in prime ideals)?
I know there still is a primary decomposition of any ideal but I don't see how to find all ideals of norm $8$ above $(2)$ in $\mathbb{Z}[\alpha]$ for example.