Questions tagged [absolute-galois-group]
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74
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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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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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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 ...
4
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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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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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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 $\mathbb{Q}$. ...
3
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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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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 ...
23
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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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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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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
...
14
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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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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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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^...
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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), ...
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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 $K$ ...
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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 ...
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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 ...
31
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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 ...
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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 ...
3
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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 ...
104
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"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 ...
8
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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 ...
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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 ...