I would like to find the minimal primary decompositions of, say, $(4)$ and $(1+\sqrt5)$ in $\mathbb{Z}[\sqrt{5}]$. In Find minimal primary decompositions of ideals there is a similar example, what troubles is that doesn't seem to exist any intuition as to why one would find that primary decomposition.
For my case:
For (4), I know that $(2,1+\sqrt{5})$ is a maximal ideal and $(2,1+\sqrt{5})^2=(4,2+2\sqrt{5})$, so I feel like this ideal will part of the primary decomposition but I can't seem to finish it.
For $(1+\sqrt{5})$, I'm also not sure what to do.