Questions tagged [absolute-galois-group]

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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 ...
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2 votes
1 answer
121 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.&...
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2 votes
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226 views

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{\...
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3 votes
0 answers
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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-...
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11 votes
0 answers
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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 ...
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3 votes
0 answers
313 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 ...
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2 votes
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231 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 ...
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1 vote
0 answers
135 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}\...
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1 vote
1 answer
233 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 ...
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8 votes
1 answer
316 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 ...
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11 votes
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218 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 ...
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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 (...
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7 votes
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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 ...
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2 votes
1 answer
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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 ...
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6 votes
1 answer
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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 ...
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7 votes
1 answer
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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 ...
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1 answer
484 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^{...
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7 votes
1 answer
728 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" ...
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7 votes
1 answer
533 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 ...
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11 votes
1 answer
809 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 $k$...
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16 votes
1 answer
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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 ...
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3 votes
1 answer
214 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 ...
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30 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 $\...
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1 vote
1 answer
157 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 ...
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3 votes
1 answer
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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 ...
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  • 189
7 votes
2 answers
1k 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 ...
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  • 1,543
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 $\...
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  • 1,543
18 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 ...
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9 votes
1 answer
365 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 ...
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11 votes
1 answer
738 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{...
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  • 1,543
2 votes
0 answers
104 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 ...
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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 ...
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  • 1,543
6 votes
2 answers
1k 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 ...
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0 votes
1 answer
200 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 ...
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  • 509
15 votes
2 answers
722 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 ...
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3 votes
0 answers
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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, ...
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2 votes
2 answers
547 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 ...
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4 votes
1 answer
142 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 $\...
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1 vote
0 answers
300 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\...
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9 votes
2 answers
948 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 ...
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  • 1,039
4 votes
0 answers
692 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 ...
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8 votes
1 answer
821 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 ...
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2 votes
2 answers
937 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 $\mathbb{Q}$. ...
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3 votes
2 answers
997 views

Is the absolute Galois group of the field of Laurent series in positive characteristic finitely generated?

If $K$ is an algebraically closed field of characteristic $p>0$, then $K((t))$, the field of Laurent series with coefficients in $K$, has infinitely many Galois extensions of degree $p$. Indeed, ...
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5 votes
2 answers
1k views

non-continuous inverse Galois problem

Let $G=Gal(\bar{\mathbf{Q}}/\mathbf{Q})$ be the absolute Galois group over $\mathbf{Q}$. Q1: Is it possible to find a (necessarily non-closed) normal subgroup $K\leq G$ such that $G/K$ is free of ...
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23 votes
3 answers
4k views

Subgroups of GL(2,q)

I have been wondering about something for a while now, and the simplest incarnation of it is the following question: Find a finite group that is not a subgroup of any $GL_2(q)$. Here, $GL_2(q)$ is ...
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1 vote
1 answer
1k views

What does Gal(Q_p/Q) mean? [closed]

What does $\mathrm{Gal}(\overline{\mathbb{Q}}_{p}/\mathbb{Q})$ mean? ($p$ is a prime number.) If it is defined as $\mathrm{Aut}(\overline{\mathbb{Q}}_{p}/\mathbb{Q})$, then does it have any property ...
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40 votes
3 answers
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On what kind of objects do the Galois groups act?

I am neither number theorist nor algebraic geometer. I am wondering whether Galois groups of number fields (say the absolute Galois group $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$) act on objects which ...
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  • 19k
14 votes
1 answer
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Is the etale fundamental group of Spec(Z)\{p_1,...,p_n} finitely presented?

(of course not, it's usually uncountable; I really mean is it the profinite completion of a finitely presented group). By definition, $\pi_1^{\operatorname{et}}(\operatorname{Spec}(\mathbb Z)\...
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29 votes
0 answers
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What do dessins tell us about the absolute Galois group?

I have sometimes seen it asserted that one manifestiation of how complicated the absolute Galois group $\mathrm{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})$ is is that one can not "pin down" any single ...
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