study of the properties of a ring with several quadratic elements

Question

here is my question. I am currently studying the case of quadratic number rings and I find quite strange that the classification of $\mathbb{Z}[\sqrt{d}]$ appears in so many books whereas I can't find any study concerning $\mathbb{Z}[\sqrt{d},\sqrt{d'}]$. For instance, what about $\mathbb{Z}[\sqrt{3},\sqrt{11}]$? It seems to me that it is noetherian, but I have no idea whether it is integrally closed or how to describe the ideals, or any other interesting property I could investigate. Thanks for any comment!

Answer 1

That is because these tend to be harder to deal with. In any case, $\Bbb Z[\sqrt3,\sqrt{11}]$ is not integrally closed, as $\alpha=\frac12(\sqrt3+\sqrt{11})$ is an algebraic integer. Indeed $\alpha^2=\frac12(7+\sqrt{33})$ so that $\alpha^4-7\alpha^2+4=0$.

But $\Bbb Z[\sqrt3,\sqrt{11}]$ is Noetherian, as it's a homomorphic image of the Noetherian domain $\Bbb Z[X,Y]$ (Hilbert Basis Theorem).