28
votes
What are dessins d'enfants?
Given a compact Riemann surface $X$, every holomorphic function on $X$ is constant. This is obvious if you think about holomorphic functions as locally conformal mappings, that is, transformations of $...
- 584
24
votes
Absolute Galois group, number theory and the Axiom of Choice
In the absence of the axiom of choice, it is still possible to define the "usual" algebraic closure of $\mathbb{Q}$ because you can just explicitly enumerate all polynomials with integer ...
- 82.7k
19
votes
Accepted
Classify all the fields with abelian absolute Galois group
Geyer in Unendliche algebraische Zahlkörper, über denen jede Gleichung auflösbar von beschränkter Stufe ist, Satz 1.13 and the paragraph after that, gives a full characterization of which abelian ...
- 2,051
17
votes
Absolute Galois group, number theory and the Axiom of Choice
There would be no consequences, for two reasons:
As Timothy Chow points out, if we define $\overline{\mathbb Q}$ as the set of complex numbers that are roots of a nonzero polynomial with rational ...
- 148k
13
votes
Teichmuller groupoids in Grothendieck's esquisse d'un programme
There is another type of important morphism between the (orbifold) fundamental groups of the moduli spaces $M_{g,\nu}\rightarrow M_{g',\nu'}$ that is considered in Grothendieck's tower. You can see ...
- 1,307
13
votes
What "should" be the absolute galois group of a field with one element
The Galois group of the maximal abelian extension of $\mathbb Q$ (or any number field) is given (class field theory) as the quotient of the idele class group by the connected component of the identity ...
- 30.5k
13
votes
Consequences of Shafarevich conjecture
The Shafarevich conjecture belongs to the broader program of Inverse Galois theory, and in that context it is just another step in that particular approach to understanding $\mathrm{Gal}(\overline{\...
- 17.6k
12
votes
Accepted
Involutions in the absolute Galois group (and the Axiom of Choice)
No, you shouldn't need any choice for this, and it should still be true if you replace $\overline{\mathbb{Q}}$ with any other algebraic closure of $\mathbb{Q}$. Let $K$ be a field (which in our ...
- 156k
10
votes
Accepted
Galois group for 0-dimensional motives
Motives of $0$-dimensional varieties are usually called Artin motives. The different fiber functors (essentially) all give rise to automorphism group isomorphic to the absolute Galois. There is one ...
- 17.4k
9
votes
Accepted
absolute Galois group of the function field of a curve over $\mathbb{F}_p^{alg}$
The theorem is true. It seems to have been proved independently by Florian Pop and by David Harbater. In Pop's paper, it is the corollary on p. 556.
MR1334484 (96k:14011)
Pop, Florian
Étale Galois ...
9
votes
Involutions in the absolute Galois group (and the Axiom of Choice)
Let me spell out a completely explicit elementary proof that visibly makes no use of choice.
Lemma 1. Let $\sigma$ be an automorphism of order $2$ of a field $K$ of characteristic $\ne2$, and let $F$ ...
- 47.1k
8
votes
Algebraically closed fields with only finite orbits
The answer is no. Let $K$ be an algebraically closed field, and let $k$ be the algebraic closure of the prime field contained in $K$. Pick a transcendence basis $B$ of $K/k$, which is nonempty if $k\...
- 28.2k
8
votes
Accepted
Galois action on automorphisms of a curve
$K$-defined would mean that the subgroup itself is defined over $K$, not the elements.
Formally, this means that there are equations over $K$ which are satisfied by the coefficients of polynomials ...
- 148k
8
votes
Accepted
Commutator subgroup of the absolute Galois group - a closed subgroup
No, the abstract commutator subgroup $[G_K,G_K]$ of the absolute Galois group $G_K$ of a number field $K$ is never closed:
Write $[G,G]$ for the commutator subgroup of $G$ as an abstract group,
and $c(...
- 2,051
7
votes
Galois group for 0-dimensional motives
What follows is a hands-on explanation (not a complete proof!) of why the Tannakian group $$G_{\textrm{dR}}(AM(\mathbb{Q}))$$ of the category of Artin motives is a nontrivial inner form of $$G_{\...
- 171
7
votes
Interesting (combinatorial) actions of the absolute Galois group $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$
You should read the paper
I. Bauer, F. Catanese, F. Grunewald: Faithful actions of the absolute Galois group on connected components of moduli spaces, Invent. Math. 199, No. 3, 859-888 (2015). ZBL1318....
- 66.3k
7
votes
Accepted
Is co-restriction in Galois cohomology in fact the norm map via Kummer isomorphism?
The corestriction map on cohomology is indeed the norm in degree zero (see Tate's notes on Galois cohomology for example). By a dimension shifting argument, it then easily follows that the ...
- 4,746
5
votes
Accepted
induced isomorphism in continuous cohomology
Here is one recent result in this direction, taken from I. Efrat and J. Minac, Galois groups and cohomological functors, Trans. of the AMS, http://www.ams.org/journals/tran/0000-000-00/S0002-9947-2016-...
- 749
4
votes
Finite-order automorphisms in the absolute Galois group of a $p$-adic field?
As Seewoo Lee points out, this is obstructed by the Artin–Schreier theorem.
Suppose that such $\sigma$ exists. By Artin–Schreier, it has order 2, and $C^\sigma \subset C$ is real-closed, and $C = C^\...
- 54.6k
4
votes
$G_{\mathbb Q}$ and primes of $\overline{\mathbb{Z}}$
There are infinitely many prime ideals $\mathfrak p$ in $\overline{\mathbf Z}$ that lie over $p$ since you can find an arbitrarily large (finite) number of prime ideals lying over $p$ in suitable ...
- 50.6k
4
votes
Accepted
Shafarevich's conjecture on Galois groups over fields ramified at finitely many places
In the footnote on Page 11 of [Fernando Q. Gouvea: Deformations of Galois Representations],
The reference is
[I.R. Shafarevich, Algebraic number fields, Proceedings of the
international Congress of ...
- 667
3
votes
Accepted
Relation between $G_{\mathbb{Q}_p}$ for different primes
As noted in the remarks above, there are different (although conjugate) embeddings $G_{\mathbb{Q}_p}\rightarrow G_\mathbb{Q}$. Let us fix one of them and denote its image by $G_p$.
If we fix prime ...
- 2,051
2
votes
Accepted
Tits algebra of the quasi-split semisimple algebraic groups
I found and read the original paper <Représentations linéaires irréductibles d'un groupe réductif sur un corps
quelconque> by Jacques Tits.
In Theorem 3.3 of this paper, it is shown that the ...
- 321
2
votes
Accepted
maximal pro-l-quotients of absolute Galois groups
I assume your field has characteristic $p>0$.
Then the maximal pro-$l$ quotient of the absolute Galois group is torsion-free. Indeed, by results of E. Becker, Euklidische Korper und euklidische ...
- 749
1
vote
Meaning of epimorphism from full Galois group to some group
If your map is continuous, its kernel is normal and closed, hence of the form $Gal(\overline{\mathbb{Q}}/F)$, where $F/\mathbb{Q}$ is finite Galois (I think you can find all the results you need and ...
- 1,766
1
vote
Recovering the Zariski topology from the Zariski topology over an extension
Let $A$ be a finitely generated $k$-algebra, and let $K$ be the algebraic
closure of $k$. I explain how to recover $spec(A)$ from $spec(A\otimes K)$.
Equivalently, I explain how to recover $spm(A)$ ...
- 76
1
vote
Consequences of Shafarevich conjecture
One additional aspect may be worth mentioning: When trying to understand the structure of an absolute Galois group ${\rm Gal}(\overline{K}/K)$ of a field $K$, it is often helpful to keep in mind its ...
- 749
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