Let $K$ be a number field, $E$ a finite extension, $A$ the integral closure of $\Bbb Z$ in $K$, $B$ the integral closure of $A$ in $E$, let $w$ be an absolute value corresponding to a prime of $B$, denote $E_w$ the completion of $E$ under $w$.
Now denote $K_w$ the closure of $K$ in $E_w$, then $EK_w$ is complete under $w$.
Why is it the case?
Since $K_w$ is the closure in $E_w$ which is complete, $K_w$ is complete. Now write $E = K(a)$, then $EK_w = K_w(a)$ is finite separable $K_w$, since $w$ is an absolute value on both $EK_w$ and $K_w$ and $K_w$ is complete, we have also $EK_w$ complete.