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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?

edited Jan 8 2012 at 19:10

asked Jan 8 2012 at 18:54

Vipul Naik
4,865●13●42

	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 2012 at 19:00
	Fixed it, thanks! – Vipul Naik Jan 8 2012 at 19:10

SGP · Accepted Answer · 2012-01-08 19:10:21Z UTC

9

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!!)

answered Jan 8 2012 at 19:10

SGP
2,614●4●14

	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 2012 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 2012 at 1:12

Ring of Witt vectors and p-adics

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