0
votes
0answers
166 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, ...
1
vote
0answers
183 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 ...
2
votes
0answers
294 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 ...
7
votes
1answer
462 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
2answers
333 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 $\mathbb{Q}$. ...
1
vote
1answer
768 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
684 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 ...
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), ...
6
votes
1answer
819 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 $K$ ...
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 ...
9
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 ...
53
votes
10answers
6k views

“Understanding” Gal(\bar Q/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 = Gal(Q bar/Q). What do people mean when they say this? The ...