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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?