Questions tagged [absolute-galois-group]
The absolute-galois-group tag has no usage guidance.
74
questions
104
votes
10
answers
17k
views
"Understanding" $\mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$
I have heard people say that a major goal of number theory is to understand the absolute Galois group of the rational numbers $G = \mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$. What do people mean when ...
55
votes
10
answers
10k
views
What are dessins d'enfants?
There was an observation that any algebraic curve over Q can be rationally mapped to P^1 without three points and this led ...
40
votes
3
answers
3k
views
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
...
31
votes
2
answers
3k
views
Number theory textbook based on the absolute Galois group?
I've just finished reading Ash and Gross's Fearless Symmetry, a wonderful little pop mathematics book on, among other things, Galois representations. The book made clear a very interesting ...
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 $\...
30
votes
0
answers
2k
views
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 ...
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 ...
22
votes
1
answer
2k
views
Are class numbers encoded in the absolute Galois group of ${\mathbb Q}$?
The absolute Galois group $G_{\mathbb Q}=\text{Gal}(\bar{\mathbb Q}/\mathbb Q)$, as a profinite group, encodes a lot of things: the whole lattice of number fields (closed subgroups of finite index), ...
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 ...
17
votes
6
answers
7k
views
Element in the absolute Galois group of the rationals
Usually when people talk on the absolute Galois group Gℚ of ℚ they have in mind two elements they can describe explicitly, namely the identity and complex conjugation (clearly, everything ...
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 ...
15
votes
1
answer
1k
views
What can we say about center of rational absolute Galois group?
Well the question is in the title.
I asked myself this question while thinking about something in Grothendieck-Teichmüller theory. I guess class field theory gives some insight into this, or I am ...
15
votes
2
answers
827
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 ...
14
votes
1
answer
1k
views
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)\...
13
votes
1
answer
1k
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 $k$...
13
votes
0
answers
235
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 ...
11
votes
1
answer
2k
views
Maximal extension almost everywhere unramified and totally split at one place
Fix a finite set of primes $S$ and an additional prime $p$. Let $K$ be the maximal extension of $\mathbb{Q}$ that is unramified outside $S$ and $\infty$ and totally split at $p$. Is the extension $K$ ...
11
votes
1
answer
819
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{...
11
votes
0
answers
235
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 ...
10
votes
0
answers
220
views
If $H$ is a quotient of $G$, does there exist an $H$-extension of $\mathbb{Q}$ not contained in a $G$-extension?
Let $\phi\colon G\rightarrow H$ be a surjective homomorphism between finite groups. Assume that $\phi$ is not split, in other words there exists no homomorphism $\sigma\colon H\rightarrow G$ such that ...
9
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 ...
9
votes
1
answer
397
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 ...
9
votes
2
answers
1k
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 ...
8
votes
1
answer
896
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 ...
8
votes
1
answer
591
views
Cartographic group and flat stringy connection
There's a literature about dessins d'enfants (including my previous question here), and one amazing thing about them is that absolute Galois group Gal Q acts on ...
8
votes
1
answer
462
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 ...
7
votes
2
answers
591
views
Involutions in the absolute Galois group (and the Axiom of Choice)
It is known that the only elementary abelian $2$-groups (finite and nonfinite) in $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ are in fact finite and cyclic – that is to say, they are of order $2$....
7
votes
2
answers
2k
views
Conjugacy classes in the absolute galois group
We consider $G_{\mathbb Q} = Gal(\mathbb {\bar Q}/\mathbb Q)$. The Frobenius elements corresponding to each prime are well-studied. But these are really not elements; these are only defined as some ...
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 ...
7
votes
1
answer
818
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" ...
7
votes
1
answer
557
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^{...
7
votes
1
answer
580
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 ...
7
votes
1
answer
744
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 ...
7
votes
1
answer
4k
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 ...
7
votes
1
answer
431
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 ...
7
votes
0
answers
263
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 ...
7
votes
0
answers
197
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 ...
7
votes
0
answers
249
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 ...
6
votes
0
answers
771
views
field automorphism action on $Ext^1(\mathbb{C}^*,\mathbb{Z})$ or $Ext^1(\bar{\mathbb Q}^\star,\mathbb{Z})$
I am interested in the action of field automorphism group $Aut(\mathbb{C}/\mathbb{Q})$ on
$Ext^1(\mathbb{C}^{\star},\mathbb{Z})$ or $Ext^1(\mathbb{\bar Q}^*,\mathbb{Z})$, and, more generally, on
$Ext^...
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 ...
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 ...
5
votes
1
answer
241
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 ...
5
votes
0
answers
222
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 ...
4
votes
1
answer
231
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 ...
4
votes
1
answer
149
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 $\...
4
votes
0
answers
790
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 ...
3
votes
1
answer
196
views
Galois action on automorphisms of a curve
Let $C$ be a smooth projective curve defined over a local field $K/\mathbb{Q}_p$. Denote by $\text{Aut}(C)$ the geometric automorphism group of $C$, which consist of isomorphisms of $C\times_{\text{...
3
votes
1
answer
223
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 ...
3
votes
1
answer
301
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 ...
3
votes
2
answers
1k
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, ...