If $K:=\mathbb{Q}(\sqrt{-5})$, find a non-trivial extension $L$ such that no prime $\mathfrak{p}\subset\mathcal{O}_K$ ramifies in $L$.
I thought about using fact that every quadratic number field is contained in a cyclotomic field, in this case $K\subset \mathbb{Q}(\zeta_{20})=\mathbb{Q}(\zeta_5)\mathbb{Q}(i)$. Since $\mathbb{Q}(\zeta_{20})|\mathbb{Q}$ has nice propreties, I suppose $L=\mathbb{Q}(\zeta_{20})$ will do the job.
The thing is: I only know how to deal with the primes $\mathfrak{p}\subset \mathcal{O}_K$ individually (finding decomposition/inertia fields etc), but I don't know how to check whether or not every $\mathfrak{p}$ is unramified.
How should I approach this?