Given $v,w$ primes of $k$, is there $K/k$ so $K_v\cap\Bbb Q^{cycl}=K_w\cap\Bbb Q^{cycl}=K\cap\Bbb Q^{cycl}$? 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.  
 A: Okay, I'm giving up.  I hadn't realized local fields had large subfields of cyclotomic extensions.  Thanks to BCnrd - I learned something interesting today.
Second try (also wrong): Choose an embedding of $k$ in an algebraic closure $\overline{\mathbb{Q}}$ of $\mathbb{Q}$, and let $k' = k \cap \mathbb{Q}^{ab}$, where $\mathbb{Q}^{ab} \subset \overline{\mathbb{Q}}$ is the maximal abelian extension.  By Kronecker-Weber, there exists a positive integer $m$ such that $k' \subset \mathbb{Q}[\zeta_m]$.  Let ${k_v}'$ be the subfield of $k_v$ consisting of elements that are algebraic over $k$ and lie in $\mathbb{Q}^{ab}$ (under any choice of embedding), and let $m_v$ be a positive integer such that ${k_v}' \subset \mathbb{Q}[\zeta_{m_v}]$.  Let $m_w$ be defined similarly.  Let $n$ be the least common multiple of $m$, $m_v$, and $m_w$, and let $K = k[\zeta_n]$.
$K$ is an abelian Galois extension of $k$ and $\mu(K) = n$.  The minimal polynomial of $\zeta_n$ over $k$ splits into (Galois conjugate) polynomials of equal degree over the subfield of $k_v \cap \overline{\mathbb{Q}}$, so $K \otimes_k k_v$ is isomorphic to a product of copies of $k_v[\zeta_n]$.  Therefore, for any place $v'$ over $v$, $K_{v'} \cong k_v[\zeta_n]$ and $\mu(K_{v'}) = n$.  Similarly, $\mu(K_{w'}) = n$ for any place $w'$ over $w$.

First Try (wrong - see BCnrd's comment): Let $n$ be the least common multiple of $\mu(k_v)$ and $\mu(k_w)$, let $K = k[\zeta_n]$, where $\zeta_n$ is a primitive $n$th root of unity, and let $v'$ and $w'$ be any chosen places over $v$ and $w$, respectively.  $K$ is Galois over $k$, and if I'm not mistaken, we have $\mu(K_{v'}) = \mu(k_v[\zeta_n]) = n$ and $\mu(K_{w'}) = \mu(k_w[\zeta_n]) = n$.  Is there a subtlety I'm missing?
