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The following corollary of Krasner´s Lemma says:

Let k be a global field and p a prime of k. Then

$(\overline{k})_p=\overline{k_p}$.

Im wondering if $(\overline{k})_p$ means the completion of $\overline{k}$ because i know that $\overline{\mathbb{Q}_p}$ is not complete. So i think it means

$\bigcup L_{ip}$ with the $L_{ip}$ ranging over all finite extensions of k.

In particular $\bigcup L_{ip}$ doesn´t need to be complete. Am i correct with that? Thx for any help given!

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You can find the corollary for example in Neukirch´s cohomology of number fields, 8.1.5.

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The equality stated in that proposition is to be read in the following way: If you take the algebraic closure of $k_p$ you obtain the same field as if you take the union $\bigcup L_{iw}$ of all completions of finite extensions $L_i$ of $k$.

This does *not* imply $\overline{k}_p$ to be complete. Actually it is false for $\overline{ \mathbb{Q}_p}$ as you mentioned.

(Look here for a short proof of the last statement: link )

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