# Examples of Completions and Algebraic Closures

It is widely known that the algebaric closure of the $p$-adic completion $\mathbb{Q}_p$ of $\mathbb{Q}$ isn't complete anymore. It's completion is complete and known as $\mathbb{C}_p$.

I have read in a book about non-archimedean analysis that in this case the process ends, which means that $\mathbb{C}_p$ is also algebraically closed.

My question is: is there an example of a field K, in which the algebraic closure $K^{alg}$ isn't complete, and the completion of $K^{alg}$ isn't algebraically closed ? And how do I construct such an example.

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Notice you can assume that $K$ is algebraically closed to begin with, and then you are looking for an algebraically closed field whose completion for an absolute value is not algebraically closed. –  Mariano Suárez-Alvarez Jan 21 '10 at 7:20

There is a theorem of Kurschák which asserts that the completion of a valued algebraically closed fied is algebraically closed. This is proved in Paulo Ribenboim's The theory of classical valuations.

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This is a special case of Zariski's lemma, which says that any finite degree extension F/K is algebraic, from which the nullstellensatz follows immediately. –  Harry Gindi Jan 21 '10 at 8:47
And of course, Zariski's lemma is a very specialized form of Zariski's Main Theorem. –  Harry Gindi Jan 21 '10 at 8:48