## Absolute Galois group of the field of Puiseux series over $\overline{\mathbb{F}}_p$

Let $K$ be the field of Puiseux series with coefficients in $\overline{\mathbb{F}}_p$ (the algebraic closure of the field with $p$ elements). What is the absolute Galois group of $K$?

Thank you to anyone who could help!

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One can compute the set of continuous homomorphisms from the absolute Galois group to $\mathbb F_p$ using the Artin-Schreier exact sequence. A basis, as a vector space over $\bar{\mathbb F}_p$ consists of functions of the form $T^{-a/b}$ with $a$, $b$, and $p$ pairwise relatively prime. This is a weird-looking countable-dimensional vector space, which makes it seem unlikely that one can find a nice description for the whole group. In particular, it cannot be finitely topologically generated. – Will Sawin Nov 16 at 6:11
Is this known if we replace $\bar{\mathbb{F}}_p$ with $\mathbb{C}$? Can one reason by the analogy? – Spice the Bird Nov 16 at 7:58
@Spice: if you replace $\overline{\mathbb{F}}_p$ iwth $\mathbb{C}$, then the field you get is algebraically closed. – Laurent Berger Nov 16 at 9:25
On a webpage entitled "Questions I'm thinking about", Kiran Kedlaya wrote "I have a method for computing in the algebraic closure of the rational function field over a finite field, using finite automata and generalized power series. Does it actually work in practice? I can't tell. (There has been a tiny bit of experimental work on this; contact me for details.)" math.ucsd.edu/~kedlaya/questions.shtml , last updated Dec. 2009. – David Speyer Nov 16 at 14:28
I think there is a 2001 paper by Kedlaya where he described the algebraic closure of $\overline{\mathbb{F}}_p((x))$ building on other's work, it is like the Puiseaux series together with towers of Artin-Schreier extensions. – 36min Nov 20 at 2:15

Let $E$ be the field $\overline{\mathbb{F}}_p((X))$. The field of Puiseux series whose exponents have denominators prime to $p$ is a subfield of $E^{sep}$, so the group you're asking about would then be the wild inertia subgroup of $Gal(E^{sep}/E)$. The group $Gal(E^{sep}/E)$ is quite complicated, and it comes up in arithmetic geometry, for example when studying the $\pi_1$ of curves. It also occurs as a closed subgroup of $Gal(\overline{\mathbb{Q}}_p / \mathbb{Q}_p)$ by the theory of the field of norms of Fontaine and Wintenberger. Its representations on $\mathbb{Z}_p$-modules are described by $\varphi$-modules'' (like $(\varphi,\Gamma)$-modules without the $\Gamma$). If you want to include Puiseux series whose exponents have denominators divisible by $p$, then you're looking at the perfection of $E$. The group does not change, as $E^{sep}$ is dense in $E^{alg}$ by a theorem of Ax.

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Thank you for the answers ! Would it be possible to explain in a few words what $(\phi, \Gamma)$-modules are ? Thanks again !

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Dear beginner, I leave to Laurent a better discussion on $(\phi,\Gamma)$-modules (although "in a few words" might be difficult even for him...), but observe that there are small boxes below answers labeled add comment which are normally used to insert these further questions or remarks. And welcome to MO, of course! – Filippo Alberto Edoardo Nov 20 at 1:05
I think that beginner has lost access to his account and his 70 points of reputation; with a new account and reputation as 1, you can neither comment nor edit your own question (since the system don't know it is yours). I remember when I was a beginner myself on this site that similar problems happened to me several times. Not sure that will help in this specific case, but here is a simple way to retrieve one's account: click to "Users" (between "Tags" and "Badge") on the top of the page, then use the search function to find your user's account, and then there should be a button saying: – Joël Nov 20 at 1:36
"is that your account?" and you're done. Anyway, welcome to MO. – Joël Nov 20 at 1:36
Thank you very much for your welcome and your explanation on how to use MO ! Joël is right: I had lost access to my account. – beginner Nov 20 at 2:38
@beginner If I can be allowed to advertize my own stuff: I wrote a survey on p-adic representations a few years ago, and one chapter concerns $(\varphi,\Gamma)$-modules. See [05] of perso.ens-lyon.fr/laurent.berger/publications.php – Laurent Berger Nov 20 at 12:27
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