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58
votes
10answers
7k 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 ...
2
votes
0answers
84 views

Galois group for 0-dimensional motives

$\newcommand{\M}{\mathcal{M}_0}$$\newcommand{\Q}{\mathbb{Q}}$ 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 ...
2
votes
2answers
349 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 $\mathbb{Q}$. ...
0
votes
0answers
193 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, ...
2
votes
2answers
402 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 ...
4
votes
1answer
111 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
2answers
943 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 ...
20
votes
3answers
1k 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, $GL_2(q)$ ...
30
votes
9answers
4k 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 ...
1
vote
0answers
189 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 ...
6
votes
2answers
577 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 ...
7
votes
1answer
479 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 ...
2
votes
0answers
325 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
2answers
315 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, ...
28
votes
3answers
2k 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 ...
1
vote
1answer
790 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 ...
11
votes
1answer
707 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 ...
4
votes
1answer
430 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 ...
11
votes
0answers
922 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 ...
6
votes
0answers
729 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 ...
18
votes
1answer
1k 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), ...
10
votes
6answers
2k 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 ...
7
votes
1answer
874 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 $K$ ...
6
votes
2answers
1k 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 ...
17
votes
2answers
2k 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 ...
2
votes
0answers
653 views

Children's drawings and Seiberg-Witten curves

This physics (bear with me for a while) paper seems to say something about Gal \bar Q/Q: Children's Drawings From Seiberg-Witten Curves, hep-th/061108. Let's ...