# Questions tagged [absolute-galois-group]

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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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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\... 2answers 917 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 ... 0answers 645 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 ... 1answer 777 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 ... 2answers 812 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$\mathbb{Q}$. ... 2answers 863 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, ... 2answers 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 ... 3answers 3k 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 ... 1answer 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 ... 3answers 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 ... 1answer 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)\...
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 ...