TITLE: Ring of Witt vectors and p-adics
QUESTION [10 upvotes]: This is probably an easy question, but I'm not able to figure it out.
Are the following the same:

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

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?

REPLY [15 votes]: 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!!)