immediate quadratic extensions to maximally complete fields - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T18:58:22Z http://mathoverflow.net/feeds/question/75137 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/75137/immediate-quadratic-extensions-to-maximally-complete-fields immediate quadratic extensions to maximally complete fields Koen S 2011-09-11T10:43:56Z 2011-09-11T10:43:56Z <p>Let $(K,\nu)$ be a field with (non-discrete) valuation. Is it possible to have a situation where $(F,\omega)$ is an immediate quadratic extension of this field with valuation such that $(F,\omega)$ is a maximally complete field?</p> <p>Some things I already know:</p> <ul> <li>the residue characteristic has to be two,</li> <li>in certain specific cases this cannot occur by results of Artin-Schreier and Kedlaya-Poonen.</li> </ul>