TITLE: Is the algebraic closure of a $p$-adic field complete
QUESTION [19 upvotes]: Let $K$ be a finite extension of $\mathbf{Q}_p$, i.e., a $p$-adic field. (Is this standard terminology?)
Why is (or why isn't) an algebraic closure $\overline{K}$ complete?
Maybe this holds more generally:
Let $K$ be a complete Hausdorff discrete valuation field. Then, why is $\overline{K}$ complete?
I think I can show that finite extensions of complete discrete valuation fields are complete.

REPLY [15 votes]: Small addition: One might try to repeat this process of algebraic closure / analytic completion. Kürschak proved that this process terminates after one more step, i.e. 
the completion of $\overline{\mathbb{Q}_p}$ is algebraically closed. 
More generally, if you start with a non-archimedean valued field $K$ and denote $\overline{K}$ the algebraic closure and $\widetilde{K}$ the analytic completion, then $\widetilde{\overline{\widetilde{K}}}$ is algebraically closed.