Questions tagged [absolute-galois-group]

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Elementary abelian 2-subgroups of $\mathrm{Aut}(\overline{\mathbb{Q}}/\mathbb{Q})$ (with and without choice)

Consider the absolute Galois group $G_{\mathbb{Q}} := \mathrm{Aut}(\overline{\mathbb{Q}}/\mathbb{Q})$. As I understand it, the only torsion elements have order $2$ (by Artin-Schreier), and they are ...
THC's user avatar
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5 votes
1 answer
220 views

Relation between $G_{\mathbb{Q}_p}$ for different primes

Let $G_{\mathbb{Q}_p}$ denote the absolute Galois group of the $p$-adic field $\mathbb{Q}_{p}$. Also, their structure as abstract groups is completely known. It is well known that this group embeds ...
kindasorta's user avatar
7 votes
0 answers
220 views

Connections between Borger's absolute geometry and Connes' and Consani's $\Gamma$-spaces

As the idea of an absolute geometry over the field with one element $\mathbb{F}_1$ becomes more clear, two approaches seem to have crystallized, being based on different assumptions and going into ...
Alexander Praehauser's user avatar
4 votes
1 answer
202 views

Shafarevich's conjecture on Galois groups over fields ramified at finitely many places

Let $K$ be a number field i.e. a finite extension of $\mathbb{Q}$, $\overline{K}$ a fixed separable closure of $K$, and $G_K:=\mathrm{Gal}(\overline{K}/K)$ the absolute Galois group of $K$. Let $S$ be ...
Nic Banks's user avatar
5 votes
0 answers
202 views

Is the group $\mathrm{Gal}(\mathbb{C}/\overline{\mathbb{Q}})$ known?

The automorphism group of the complex numbers $\mathbb{C}$ and the Galois group $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ are amongst the most mysterious and worst understood objects in Galois ...
THC's user avatar
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5 votes
2 answers
1k views

Absolute Galois group, number theory and the Axiom of Choice

Richard Taylor once explained his research (in number theory) in a very simple way: understanding the absolute Galois group $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$. It is known that in ...
THC's user avatar
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3 votes
0 answers
178 views

Dessins d'enfants and the absolute Galois group

If I am not mistaken, Alexander Grothendieck introduced dessins d'enfants as they are known today, in order to better understand the absolute Galois group $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}...
THC's user avatar
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6 votes
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189 views

Reference request: projectivity of the absolute Galois group of $\mathbb{Q}^{\mathrm{ab}}$

$\newcommand{\ab}{\mathrm{ab}}$Let $\mathbb{Q}^{\ab}$ denote the maximal abelian extension of $\mathbb{Q}$. I have heard the absolute Galois group of $\mathbb{Q}^{\ab}$ is projective (e.g. see this ...
projectivityq's user avatar
2 votes
1 answer
128 views

Tits algebra of the quasi-split semisimple algebraic groups

My question arose from the proof of Proposition 31.7 of "The book of involutions." It says "… is the Tits algebra of the quasisplit group $(G_{\nu_G})_{F_{\chi}}$, hence it is trivial.&...
YJ Kim's user avatar
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Is the absolute Galois group $\text{Gal}(\bar K/K)$ isomorphic to $\text{Gal}(K(S)/K)$?

Let $K$ be a finite extension of $\mathbb{Q}_p$ with ring of integers $\mathcal{O}$, maximal ideal $\mathfrak{m}$ and uniformizer $\pi$. Let $\bar K$ be the algebraic closure of $K$ and $\bar{\...
MAS's user avatar
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The Brauer group and the second Galois cohomology group

I know that for any finite Galois extension K of $\mathbb{Q}$ the group $H^2(Gal(K/\mathbb{Q}),K^*)$ is isomorphic to the Brauer group $Br(K/\mathbb{Q})$. The isomorphism goes as follows: to a 2-...
Jacques's user avatar
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13 votes
0 answers
221 views

Dessins d'enfant of Dynkin diagrams?

Dessins d'enfant have a nice particular case of Shabat trees, where we take a tree, bicolor it, and get a polynomial map. A very famous set of trees are the Dynkin diagrams. I wonder what are the ...
Andy's user avatar
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356 views

How are problems about number fields reduced to problems about their absolute Galois groups?

The article on Wikipedia about Neukirch–Uchida theorem claims right from the beginning the statement in my question. I have seen similar claims elsewhere before. I am a little puzzled by this ...
Yujia Yin's user avatar
2 votes
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251 views

Absolute Galois group of Q and stratification of moduli space of curves

This is slightly related, but distinct from, a question I asked earlier. The moduli space of ribbon graphs with metric (with all vertices having degree at least 3) is isomorphic to the moduli space of ...
Waleed Qaisar's user avatar
1 vote
0 answers
180 views

Is semi-simplicity of Galois representations local?

Let $\rho:G_{\mathbb{Q}}\rightarrow \text{Gl}(V)$ be a finite dimensional $\ell$-adic Galois representation. Then for each prime, by pre-composing $\rho$ with the natural inclusion $G_{\mathbb{Q}_p}\...
curious math guy's user avatar
1 vote
1 answer
324 views

$G_{\mathbb Q}$ and primes of $\overline{\mathbb{Z}}$

I know that if $K/\mathbb Q$ is a finite Galois extension (i.e. a Galois number field), then for any prime $(p)\subseteq \mathbb Z$, the Galois group $G=\operatorname{Gal}(K/\mathbb Q)$ acts ...
Gaurav Goel's user avatar
8 votes
1 answer
410 views

Commutator subgroup of the absolute Galois group - a closed subgroup

Let $K$ be a finite extension of $\mathbb{Q}$. Is it possible that the commutator subgroup of the absolute Galois group of $K$ (considered as an abstract group) is a closed subgroup? This property ...
user avatar
11 votes
0 answers
227 views

Grothendieck-Teichmüller conjecture and tropicalization of moduli of curves

Abramovich, Caporaso and Payne (2014) have constructed functorial tropicalization maps from the Berkovich analytification of the moduli spaces of stable curves, $\overline{M}_{g,n}$, to the moduli ...
user138464's user avatar
1 vote
2 answers
278 views

Meaning of epimorphism from full Galois group to some group

My problem has two parts: let $\;G:=\operatorname{Gal}(\overline{\Bbb Q}/\Bbb Q)\;$ be the full Galois group of the rationals and $\;K\;$ be some finite group, then: (1) Does having an epimorphism (...
Ninet's user avatar
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0 answers
235 views

Is this Related to Tannakian Formalism?

I am wondering how I might be able to express the following phenomenon, which is essentially equivalent to Artin's linear independence of characters, in Tannakian formalism. Any help would be much ...
Cayley-Hamilton's user avatar
2 votes
1 answer
238 views

maximal pro-l-quotients of absolute Galois groups

Let $K$ be a field, preferably a function field of a variety $X$ over $\overline{\mathbb{F}}_p$. I am looking for an answer or existing literature on the following question: What is known about the ...
darko's user avatar
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7 votes
1 answer
546 views

Interesting (combinatorial) actions of the absolute Galois group $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$

I have read many times that it is crucial to understand the absolute Galois group $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) =: L$, in so far that some have stated (was it Richard Taylor ?) that ...
THC's user avatar
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7 votes
1 answer
425 views

Recovering the Zariski topology from the Zariski topology over an extension

Suppose $A$ is a $k$-algebra, with $k$ a field, and let $\ell$ be a field extension of $k$. Is there an easy way to see/recover $\mathrm{Spec}(A)$ in/from $\mathrm{Spec}(A \otimes_k \ell)$, using the ...
THC's user avatar
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7 votes
1 answer
519 views

absolute Galois group of the function field of a curve over $\mathbb{F}_p^{alg}$

In their 2008 paper "Torelli theorem for curves over finite fields" Bogomolov, Korotiaev and Tschinkel mention in the beginning of Section 9 that absolute Galois groups of curves over $\mathbb{F}_p^{...
Dima Sustretov's user avatar
7 votes
1 answer
773 views

Teichmuller groupoids in Grothendieck's esquisse d'un programme

Grothendieck in his Esquisse d'un programme mentioned without any precise definition and construction that the absolute Galois group $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$ acts on the whole "tower" ...
asv's user avatar
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7 votes
1 answer
649 views

Is co-restriction in Galois cohomology in fact the norm map via Kummer isomorphism?

Let $\mathrm{F}$ be a field that contains a root of unity of order $p$, where $p$ is a prime number. Fix an element $a$ such that $a \in \mathrm{F}$ and $\sqrt[p]{a} \notin \mathrm{F}$. Consider the ...
Guy's user avatar
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11 votes
1 answer
989 views

Classify all the fields with abelian absolute Galois group

I'm wondering if anyone has classified all the fields $K$ such that $Gal(\bar{K}/K)$ is abelian? The only examples I'm aware of are: finite fields, the real numbers $\mathbb{R}$ and $k((T))$ where $k$...
S. Li's user avatar
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16 votes
1 answer
1k views

What "should" be the absolute galois group of a field with one element

As far as I know there is many "suggestions" of what should be a "field with one element" $\mathbf{F}_{1}$. My question is the following: How we should think or what should be the "absolute Galois ...
Muhammed Ali's user avatar
3 votes
1 answer
216 views

induced isomorphism in continuous cohomology

Suppose that we have a morphism between profinite groups $f: G_{1}\rightarrow G_{2}$ such that $f^{\ast}:H_{cont}^{\ast}(G_{2},A)\rightarrow H_{cont}^{\ast}(G_{1},A) $ is an isomorphism for any finite ...
Muhammed Ali's user avatar
31 votes
3 answers
1k views

Consequences of Shafarevich conjecture

The Shafarevich conjecture states that the Galois group $\mathrm{Gal}({\overline{\mathbf{Q}}/\mathbf{Q}^{ab}})$ is a free profinite group, where $\mathbf{Q}^{ab}$ is the maximal abelian extension of $\...
Muhammed Ali's user avatar
1 vote
1 answer
160 views

Another fix field of a certain galois group action

Let $E=\mathbb{F}_p(\!(u)\!)$, the Laurent series field over $\mathbb{F}_p$. Let $K/E$ be a finite normal separable extension. Consider the field $L=K(x \mid x^p-x-a=0 \text{ for some } a \in K)$. Let ...
Louis's user avatar
  • 189
3 votes
1 answer
273 views

Fix field of a certain galois group action

Let $E= \mathbb{F}_p(\!(u)\!)$, $E^s$ a separable closure of $E$ and write $G_E= \mathrm{Gal}(E^s/E)$ for the absolute Galois group of $E$. Take a lift of the $u$-adic valuation on $E$ to $E^s$ and ...
Louis's user avatar
  • 189
8 votes
2 answers
2k views

Langlands program vs Shimura-Taniyama-Weil conjecture

Edward Frenkel said that "we can see Langlands program as a generalization of Shimura-Taniyama-Weil conjecture in the case of elliptic curves" I hope I'm not distorting his phrase, can someone ...
Ofra's user avatar
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1 vote
2 answers
1k views

Is the absolute Galois group the same as the automorphism group? [closed]

Is the absolute Galois group $\mathrm{Gal}(\overline{\mathbf{Q}}|\mathbf{Q})$ the same as the group $\mathrm{Aut}_{\mathbf{Q}}(\overline{\mathbf{Q}})$ the automorphism group in the category of $\...
Ofra's user avatar
  • 1,573
19 votes
3 answers
2k views

Profinite groups as absolute Galois groups

It is a well-known result that all profinite groups arise as the Galois group of some field extension. What profinite groups are the absolute Galois group $\mathrm{Gal}(\overline{K}|K)$ of some ...
Myshkin's user avatar
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9 votes
1 answer
383 views

Weight filtration on certain Galois representations

Let $G$ be the absolute Galois group of a number field $K$. Let $\ell$ be a prime number. There are representations $\mathbb{Z}_\ell(n)$ of $G$ on the group of $\ell$-adic integers given by the ...
Geoffroy Horel's user avatar
11 votes
1 answer
777 views

Dessins d'enfants and absolute Galois group

I would like to know what is the recent progress about the group homomorphism $$ \mathrm{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})\rightarrow \mathrm{Out}(\hat{F_{2}})$$ $\mathrm{Gal}(\overline{\mathbf{...
Ofra's user avatar
  • 1,573
2 votes
0 answers
106 views

Which fields have no extensions of degree divisble by a fixed prime?

Let $p$ be a prime. What are the most general examples of a field $K$ such that for any finite extension $L/K$ the degree $[L:K]$ is prime to $p$? Certainly, there are algebraically closed examples ...
Mikhail Bondarko's user avatar
7 votes
1 answer
3k views

Unramified extension of number fields

Any finite field extension (in particular Galois extension) of $\mathbb{Q}$ is ramified. Is there an intuitive geometric explanation of this fact? Suppose we have an number field $K$, is any Galois ...
Ofra's user avatar
  • 1,573
7 votes
2 answers
2k views

Frobenius elements in infinite extensions

Let $K$ be a number field, $\bar K$ an algebraic closure and $G$ the associated absolute Galois group. How can I define the Frobenius elements of $G$ or at least their conjugacy class? I know how ...
user81319's user avatar
0 votes
1 answer
210 views

Fixed field of the Nebentypus of a newform for $\Gamma_1(N)$

Let $f=\sum_{n\geq 1}\in S_2(\Gamma_1(N),\varepsilon)$ be a normalized newform without CM and with Nebentypus $\varepsilon$. Let $L=\mathbb Q(a_n\colon n\in \mathbb N)$ be the number field generated ...
Ferra's user avatar
  • 509
15 votes
2 answers
785 views

Galois group for 0-dimensional motives

It is my understanding that in dimension 0, the theory of motives should just be Galois theory for fields. I am hoping to find a reference or two to help me get some things straightened out. One can ...
Julian Rosen's user avatar
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3 votes
0 answers
1k views

English version of "Quasi-Hopf Algebras"

I was wondering where I can find a pdf of Drinfeld's paper "Quasi-Hopf Algebras," which formulated the Grothendieck-Teichmuller group. The Russian version is in Algebra i Analiz, 1:6 (1989), 114–148, ...
Ravi Jagadeesan's user avatar
2 votes
2 answers
553 views

The relationship between SL(2,Z) and Gal(Qbar,Q)

(caveat: I'm not a number-theorist or Langlands-programme-er, and I don't expect to understand all the answers to this question, but I figured they might be useful to someone besides me). I've been ...
graveolensa's user avatar
4 votes
1 answer
146 views

Galois action on ultrapowers

Let $K$ be a char $0$ field with algebraic closure $\bar K$ and absolute Galois group $G$. Let $\mathcal U$ be an ultrafilter on $\mathbb N$ and $F=\bar K^\mathbb N/\mathcal U$ be the ultrapower of $\...
Cyrille Corpet's user avatar
1 vote
0 answers
308 views

Extending systems of l-adic representations to other l

I'm asking this not because I have an idea how one might approach it, but because it seems natural and inherently interesting. Let $K$ be a number field, $G_K$ its absolute Galois group, and $\ell\...
David Corwin's user avatar
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9 votes
2 answers
975 views

Is it known if the absolute Galois group is "divisible"?

The definitions of a divisible group that I have seen all seem to assume abelian is an a priori property of the group. My question is as to whether or not it is known that--given a non-torsion element ...
Adam Hughes's user avatar
  • 1,029
4 votes
0 answers
740 views

Haar measure on Galois groups

Galois groups are nice compact Hausdorff groups, and therefore possess a bounded Haar measure, unique if we insist that the total volume be $1$. What is the Haar measure on the absolute Galois group ...
Filippo Alberto Edoardo's user avatar
8 votes
1 answer
868 views

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 ...
beginner's user avatar
2 votes
2 answers
1k views

Place stabilizers for the absolute Galois Group

Fix an algebraic closure, $\overline{\mathbb{Q}}$ for the rationals and consider the set, $B_p$, of all places of $\overline{\mathbb{Q}}$ over a fixed (possibly infinite) prime, $p$, of $\mathbb{Q}$. ...
Adam Hughes's user avatar
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