= Solution
The <completion of a number field at a prime ideal> does not change either invariant. More precisely, $\widehat B_q$ is the <valuation ring> of $L_q$, its maximal ideal is $q\widehat B_q$, and
$$
\boxed{e(L_q/K_p)=e_{q/p},\qquad
\widehat B_q/q\widehat B_q\cong B/q.}
$$
Thus the completed extension has the same <ramification index> and the same <residue-field extension>, hence the same <residue-field degree> $f_{q/p}$. This is <completion preserves ramification and residue data>.
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