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.

  • $\begingroup$ Does that help? What do you mean? $\endgroup$ – Rubisk Jan 16 at 19:01
  • $\begingroup$ 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). $\endgroup$ – Rubisk Jan 19 at 21:38
  • $\begingroup$ Assuming the class group is trivial? $\endgroup$ – Rubisk Jan 19 at 21:57

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.