# Consider the order $\mathbb{Z}[\sqrt[4]{24}]$. Find all ideals of norm 100.

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.

• Does that help? What do you mean? – Rubisk Jan 16 at 19:01
• Why? Couldn't an ideal of norm 100 contract to an ideal of norm 25 or something? And I am talking about the non-maximal order $\mathbf{Z}[\sqrt[4]{24}]$ (the cuberoot is a typo I dont know how to fix). – Rubisk Jan 19 at 21:38
• Assuming the class group is trivial? – Rubisk Jan 19 at 21:57