I'm having some problems finding the ring of integers of $\mathbb{Q}(\sqrt{-3},\sqrt{5})|\mathbb{Q}$. How can I find it?
Also, I'd like to prove that $\alpha:=\frac{1+\sqrt{-3}+\sqrt{5}+\sqrt{-15}}{4}$ generates a subgroup of finite index of the group of units of $\mathbb{Q}(\sqrt{-3},\sqrt{5})|\mathbb{Q}$, but I don't know how to apply the Dirichlet theorem to prove it (that's the only point I've got to so far).
I already know how to compute its discriminant and therefore I know which primes ramify, the problem is that I really don't know how to find its ring of integers, even though I've felt the temptation of writting $\mathcal{O}_{\mathbb{Q}(\sqrt{-3},\sqrt{5})}=\mathbb{Z}\left[\frac{1+\sqrt{-3}}{2},\frac{1+\sqrt{5}}{2}\right]$
Thanks in advance.