For any field $k$, let $\mu(k)$ denote the roots of unity in $k$. Now let $k$ be a number field and let $v, w$ be non-archimedean primes of $k$ with distinct residual characteristics. Does there exist a finite Galois extension $K/k$, with $v',w'$ primes of $K$ lying over $v,w$, such that $\mu(K)=\mu(K_{v'})=\mu(K_{w'})$?

For example, if $k=\mathbb{Q}$ and you're looking at the primes $3$ and $5$, then you can take $K=\mathbb{Q}(\zeta_{24})$ as your galois extension.