Examples of Completions and Algebraic Closures - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T08:16:50Z http://mathoverflow.net/feeds/question/12508 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/12508/examples-of-completions-and-algebraic-closures Examples of Completions and Algebraic Closures Willem Noorduin 2010-01-21T07:03:58Z 2010-01-21T13:18:12Z <p>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\$. </p> <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.</p> <p>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.</p> http://mathoverflow.net/questions/12508/examples-of-completions-and-algebraic-closures/12509#12509 Answer by Mariano Suárez-Alvarez for Examples of Completions and Algebraic Closures Mariano Suárez-Alvarez 2010-01-21T07:29:24Z 2010-01-21T07:34:35Z <p>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 <em>The theory of classical valuations</em>.</p> http://mathoverflow.net/questions/12508/examples-of-completions-and-algebraic-closures/12510#12510 Answer by Paul Ziegler for Examples of Completions and Algebraic Closures Paul Ziegler 2010-01-21T07:32:05Z 2010-01-21T07:32:05Z <p>No, there is not.</p> <p>If the valuation is archimedean, by Ostrowski the field is isomorphic to the real or complex numbers, so the algebraic closure will already be complete.</p> <p>If the valuation is non-archimedean, the completion of the algebraic closure will always be algebraically closed. See for example here: <a href="http://math.stanford.edu/~conrad/248APage/handouts/algclosurecomp.pdf" rel="nofollow">http://math.stanford.edu/~conrad/248APage/handouts/algclosurecomp.pdf</a></p>