The number fields are $K=\mathbb{Q}(\sqrt{10})$ and $L=K(\sqrt{-p})=\mathbb{Q}(\sqrt{10},\sqrt{-p})=\mathbb{Q}[x]/\left(x^4-(20-2p)x^2+(p+10)^2\right)$ where I already proved the last equality where the polynomial has the root $\sqrt{10}+\sqrt{-p}$.
The problem is to show that $[\mathscr{O}_L^*:\mathscr{O}_K^*]=3,1$ for $p=3$ or $p\neq3$ respectively.
What I have shown is that $\eta=3+\sqrt{10}$ is a fundamental unit, so that $\mathscr{O}_K^*=\langle-1,\eta\rangle$. I think that first the finiteness of the norm has to be shown and that can be done with reasoning about them (both unit groups of the ring of integers of $K$ and $L$) both having a finite basis(??). Then maybe show that it divides a multiple of $3$ and then saying something about the norm of $\mathscr{O}_L^*$ over $L/\mathbb{Q}$. Should I try to compute $\mathscr{O}_L$ or is its unit group way easier to determine, like $\{\pm1\}$ or something? Literally clueless on how to attack this problem.