Ring of Witt vectors and p-adics - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T19:55:19Z http://mathoverflow.net/feeds/question/85204 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/85204/ring-of-witt-vectors-and-p-adics Ring of Witt vectors and p-adics Vipul Naik 2012-01-08T18:54:57Z 2012-01-08T19:10:21Z <p>This is probably an easy question, but I'm not able to figure it out.</p> <p>Are the following the same:</p> <ol> <li><p>Field of fractions of the ring of Witt vectors over the algebraic closure of $\mathbb{F}_p$</p></li> <li><p>Algebraic closure of the field of p-adics (which is the field of fractions of the ring of Witt vectors over $\mathbb{F}_p$)</p></li> </ol> <p>In other words, does the operation of taking the field of fractions of the ring of Witt vectors commute with the operation of taking the algebraic closure?</p> http://mathoverflow.net/questions/85204/ring-of-witt-vectors-and-p-adics/85205#85205 Answer by SGP for Ring of Witt vectors and p-adics SGP 2012-01-08T19:10:21Z 2012-01-08T19:10:21Z <p>no: the Witt ring of $\bar{F_p}$ is a complete DVR and so its field of fractions will be a complete local field; but the algebraic closure of $Q_p$ is not complete. </p> <p>However, take the maximal unramified extension of $Q_p$; this is a non-complete field. Its completion $F$ is the fraction field of the Witt ring of $\bar{F_p}$ and the Witt ring itself is the ring of integers in $F$. </p> <p>(Serre's Local Fields contains all of this and much more!!)</p>