10
$\begingroup$

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?

$\endgroup$
0

1 Answer 1

16
$\begingroup$

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!!)

$\endgroup$
2
  • 4
    $\begingroup$ 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. $\endgroup$
    – Lubin
    Jan 9, 2012 at 20:31
  • $\begingroup$ @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!! $\endgroup$
    – SGP
    Jan 14, 2012 at 1:12

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.