Let $p_i$ be distinct primes. Denote the integral closure of $\mathbb Q(\sqrt{p_1},\sqrt{p_2},\dots)$ by $A$.
Is $A$ noetherian or artinian?
For number field, the artinian part is easy as you have DVR and 1-1 correspondence of primes with localization. I think it is not artinian by considering $(\sqrt{\prod_{i\leq n}p_i})$ forming descending chain. I could not construct an obvious ascending sequence. I could try take $A_i$ be integral closure of $Q(\sqrt{p_1},\dots,\sqrt{p_i})$ and use ramification theory of $p_i$ here. However the ring of coefficient changes and I need to extend the ramified ideals to $A$ which does not guarantee the chain is proper ascending.