Let $L$ be algebraic extension of $\Bbb{Q}_p$ and $α∈\overline{\Bbb{Q}_p}$. Let $ \widehat L$ be completion of $L$. Is $ \widehat L(α)$ is $ \widehat {L(α)}$ ?
It plausibly true, but I cannot prove this. Perhaps there maybe counterexamples.
Thank you for your help.