I am struggling with how to find a principal element for each of $(2, 1 + \sqrt[3]{5})$ and $(2, \sqrt[3]{25} - \sqrt[3]{5} + 1)$ in $\mathbb{Z}[\sqrt[3]{5}]$. Working with norms here seems ugly. The main aim is to prove it is a PID via factoring $(2)$, $(3)$, $(5)$, $(7)$ and the Minkowski bound.
Any help appreciated!