All Questions
Tagged with galois-theory profinite-groups
16 questions
27
votes
1
answer
2k
views
Galois Group as a Sheaf
I've noticed that the Galois groups associated to Galois field extensions $L$ of a given field $K$ seem remarkably like a sheaf, with the field extensions taking the place of open set, and the Galois ...
- 15.4k
25
votes
2
answers
2k
views
Profinite groups as étale fundamental groups
Does every profinite group arise as the étale fundamental group of a connected scheme?
Equivalently, does every Galois category arise as the category of finite étale covers of a connected scheme?
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- 63.1k
17
votes
3
answers
2k
views
Finitely generated Galois groups
It is well-known that for a given natural number $n$ there is only finite number of extensions of $\mathbb Q_p$ of degree $n$. This result appears in many introductory books on algebraic number theory....
- 1,418
15
votes
0
answers
458
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Is the absolute Galois group of the rationals Hopfian?
Is every continuous epimorphism from the absolute Galois group of $\mathbb{Q}$ to itself injective?
- 11.3k
8
votes
3
answers
932
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Centralizers of elements in free profinite groups
I believe, although I can't say that I've given a rigorous proof, that for a free group $F_r$, and an element of it $a$, $C_{F_r}(\langle a \rangle)=$ the group generated by the elements $b \in F_r$ ...
- 5,929
5
votes
1
answer
496
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“Sheaf cohomology” of Galois groups
Apparently the Galois group $G$ of a Galois extension $E/F$ can be viewed as an “étale sheaf” on the set $X$ of intermediate Galois extensions equipped with an appropriate Grothendieck topology (see ...
- 531
5
votes
1
answer
314
views
abelian and nonabelian parts of Aut($\widehat{F_2}$)
Let $F$ be the free profinite group on two generators. Let $\text{IA}(F) := \ker\left(\text{Aut}(F)\rightarrow GL_2(\widehat{\mathbb{Z}})\right)$, the group of "IA automorphisms" of $F$. (I'm also ...
- 10.7k
5
votes
0
answers
149
views
Can there be non-isomorphic fundamental groups of equivalent Galois categories?
It is known that if $(C, F)$ is a Galois category then there exists an equivalence $C \cong \pi_1(C, F)-FinSets$ between $C$ and the category of finite sets with continuous actions of $\pi_1(C, F) := ...
- 1,516
5
votes
0
answers
245
views
Why are procyclic subgroups of Galois groups of number fields free profinite?
On p832 of Coombes, Harbater - Hurwitz familes and arithmetic Galois groups, the following is claimed:
Let $K$ be a number field, take $1 \neq \omega \in \mathrm{Gal}(\bar{\mathbb{Q}}/K)$, and let $...
- 1,338
4
votes
2
answers
232
views
What is the probability of generating a given procyclic subgroup in $\mathrm{Gal}(\bar{K}/K)$?
This question began as Why are procyclic subgroups of Galois groups of number fields free profinite?, which fizzled out, but which garnered some helpful comments from YCor.
Let $K$ be a field, take $\...
- 1,338
4
votes
0
answers
145
views
Proving that the fundamental group of a finite Galois category is profinite
This is sort of a cross-posting of this question of mine over on math.SE.
Suppose that I am given a finite Galois category $(\mathcal{G}, F)$, i.e. a Galois category in the sense of SGA 1. One of the ...
- 1,516
3
votes
0
answers
89
views
Which elements in $\mathrm{Aut}(\widehat{F_2})$ preserve the procyclic subgroup generated by the commutator $c=[a,b]$?
Let $F_2$ denote the free group over two generators $a,b$, and we denote $c=[a,b]$ as the commutator. It is well-known that any automorphism $\psi$ of $F_2$ preserves the conjugacy class of the ...
- 605
3
votes
0
answers
383
views
Semidirect product in inverse Galois problem
Let $L/\mathbb{Q}$ (resp. $K/\mathbb{Q}$) be a Galois extension of rational number field $\mathbb{Q}$ with Galois group $P$ (resp. $H$) where $P$ is a second countable pro-$p$ group and $H$ is a ...
- 667
3
votes
0
answers
242
views
Finitely generated subgroups of the absolute Galois group
Consider the absolute Galois group $\operatorname{Gal}(\overline{\mathbb{Q}} / \mathbb{Q})$. It seems to me that, in general, providing an "explicit" description of the elements of this ...
2
votes
0
answers
160
views
Problem with a proof of Wilson's 'Profinite groups'
(Crossposted on StackExchange Mathematics: https://math.stackexchange.com/questions/2391626/problem-with-a-proof-of-wilsons-profinite-groups)
I need help with the proof of Proposition (3.1.3) given ...
- 41
1
vote
2
answers
1k
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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 $\...
- 1,613