This is probably an easy question, but I'm not able to figure it out.


Are the following the same:


1. Field of fractions of the ring of Witt vectors over the algebraic closure of $\mathbb{F}\_p$
2. Algebraic closure of the field of p-adics (which is the field of fractions of the ring of Witt vectors over $\mathbb{F}\_p$)


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?



Ring of Witt vectors and p-adics
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. 


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 $F$. 


(Serre's Local Fields contains all of this and much more!!)


