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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?

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I assume "Witt ring" means "ring of Witt vectors". Probably worth a mention, since Witt ring normally means something very different (unfortunately). –  darij grinberg Jan 8 '12 at 19:00
    
Fixed it, thanks! –  Vipul Naik Jan 8 '12 at 19:10

1 Answer 1

up vote 10 down vote accepted

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!!)

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Indeed, one should think of the Witt construction as a machine for producing unramified extensions of ${\mathbb{Q}}_p$. Since ${\mathbb{Q}}_p$ has plenty of ramified extensions, the Witt construction will always give you something far from the alg. closure. –  Lubin Jan 9 '12 at 20:31
    
@Lubin:Indeed! Thanks! I forgot to mention that the Witt ring of $\bar{F_p}$ plays an important role in the theory of Lubin-Tate formal groups!! –  SGP Jan 14 '12 at 1:12

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